We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See (Figure). Show
Figure 2. We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write[latex]\,\left(0,\text{ }100\right].\,[/latex]We will discuss interval notation in greater detail later. Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative. Before we begin, let us review the conventions of interval notation:
See (Figure) for a summary of interval notation. Figure 3. Finding the Domain of a Function as a Set of Ordered PairsFind the domain of the following function:[latex]\,\left\{\left(2,\text{ }10\right),\left(3,\text{ }10\right),\left(4,\text{ }20\right),\left(5,\text{ }30\right),\left(6,\text{ }40\right)\right\}[/latex]. Show Solution First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs. [latex]\left\{2,3,4,5,6\right\}[/latex] Try ItFind the domain of the function: [latex]\left\{\left(-5,4\right),\left(0,0\right),\left(5,-4\right),\left(10,-8\right),\left(15,-12\right)\right\}[/latex] Show Solution [latex]\left\{-5,\,0,\,5,\,10,\,15\right\}[/latex] How ToGiven a function written in equation form, find the domain.
Finding the Domain of a FunctionFind the domain of the function[latex]\,f\left(x\right)={x}^{2}-1.[/latex] Show Solution The input value, shown by the variable[latex]\,x\,[/latex]in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers. In interval form, the domain of[latex]\,f\,[/latex]is[latex]\,\left(-\infty ,\infty \right).[/latex] Try ItFind the domain of the function:[latex]\,f\left(x\right)=5-x+{x}^{3}.[/latex] Show Solution [latex]\left(-\infty ,\infty \right)[/latex] How ToGiven a function written in an equation form that includes a fraction, find the domain.
Finding the Domain of a Function Involving a DenominatorFind the domain of the function[latex]\,f\left(x\right)=\frac{x+1}{2-x}.[/latex] Show Solution When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 2-x& =& 0\hfill \\ \hfill -x& =& -2\hfill \\ \hfill x& =& 2\hfill \end{array}[/latex] Now, we will exclude 2 from the domain. The answers are all real numbers where[latex]\,x<2\,[/latex]or[latex]\,x>2\,[/latex]as shown in (Figure). We can use a symbol known as the union,[latex]\,\cup ,[/latex]to combine the two sets. In interval notation, we write the solution:[latex]\left(\mathrm{-\infty },2\right)\cup \left(2,\infty \right).[/latex] Figure 4. Try ItFind the domain of the function:[latex]\,f\left(x\right)=\frac{1+4x}{2x-1}.[/latex] Show Solution [latex]\left(-\infty ,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)[/latex] How ToGiven a function written in equation form including an even root, find the domain.
Finding the Domain of a Function with an Even RootFind the domain of the function[latex]\,f\left(x\right)=\sqrt{7-x}.[/latex] Show Solution When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 7-x& \ge & 0\hfill \\ \hfill -x& \ge & -7\hfill \\ \hfill x& \le & 7\hfill \end{array}[/latex] Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to[latex]\,7,\,[/latex]or[latex]\,\left(-\infty ,7\right].[/latex] Try ItFind the domain of the function[latex]\,f\left(x\right)=\sqrt{5+2x}.[/latex] Show Solution [latex]\left[-\frac{5}{2},\infty \right)[/latex] Can there be functions in which the domain and range do not intersect at all? Yes. For example, the function[latex]\,f\left(x\right)=-\frac{1}{\sqrt{x}}\,[/latex]has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common. Using Notations to Specify Domain and RangeIn the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example,[latex]\,\left\{x|10\le x<30\right\}\,[/latex]describes the behavior of[latex]\,x\,[/latex]in set-builder notation. The braces[latex]\,\left\{\right\}\,[/latex]are read as “the set of,” and the vertical bar | is read as “such that,” so we would read[latex]\,\left\{x|10\le x<30\right\}\,[/latex]as “the set of x-values such that 10 is less than or equal to[latex]\,x,\,[/latex]and[latex]\,x\,[/latex]is less than 30.” (Figure) compares inequality notation, set-builder notation, and interval notation. Figure 5. To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol,[latex]\,\cup ,[/latex]to combine two unconnected intervals. For example, the union of the sets[latex]\left\{2,3,5\right\}\,[/latex] [latex]\left\{x|\text{ }|x|\ge 3\right\}=\left(-\infty ,-3\right]\cup \left[3,\infty \right)[/latex] Set-Builder Notation and Interval NotationSet-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form[latex]\left\{x|\,\text{statement about }x\right\}\,[/latex]which is read as, “the set of all[latex]\,x\,[/latex]such that the statement about[latex]\,x\,[/latex]is true.” For example, [latex]\left\{x|4<x\le 12\right\}[/latex] Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example, [latex]\left(4,12\right][/latex] How ToGiven a line graph, describe the set of values using interval notation.
Describing Sets on the Real-Number LineDescribe the intervals of values shown in (Figure) using inequality notation, set-builder notation, and interval notation. Figure 6. Show Solution To describe the values,[latex]\,x,\,[/latex]included in the intervals shown, we would say, “[latex]x\,[/latex]is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.” Inequality[latex]1\le x\le 3\,\text{or}\,x>5[/latex]Set-builder notation[latex]\left\{x|1\le x\le 3\,\text{or}\,x>5\right\}[/latex]Interval notation[latex]\left[1,3\right]\cup \left(5,\infty \right)[/latex]Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set. Try ItGiven (Figure), specify the graphed set in
Figure 7.
Finding Domain and Range from GraphsAnother way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See (Figure). Figure 8. We can observe that the graph extends horizontally from[latex]\,-5\,[/latex]to the right without bound, so the domain is[latex]\,\left[-5,\infty \right).\,\,[/latex]The vertical extent of the graph is all range values[latex]\,5\,[/latex]and below, so the range is[latex]\,\left(\mathrm{-\infty },5\right].\,[/latex]Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range. Finding Domain and Range from a GraphFind the domain and range of the function[latex]\,f\,[/latex] Figure 9. Show Solution We can observe that the horizontal extent of the graph is –3 to 1, so the domain of[latex]\,f\,[/latex] The vertical extent of the graph is 0 to –4, so the range is[latex]\,\left[-4,0\right).\,[/latex]See (Figure). Figure 10. Finding Domain and Range from a Graph of Oil ProductionFind the domain and range of the function[latex]\,f\,[/latex]whose graph is shown in (Figure). Figure 11. (credit: modification of work by the U.S. Energy Information Administration) [2] Show Solution The input quantity along the horizontal axis is “years,” which we represent with the variable[latex]\,t\,[/latex]for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable[latex]\,b\,[/latex]for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as[latex]\,1973\le t\le 2008\,[/latex]and the range as approximately[latex]\,180\le b\le 2010.[/latex] In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines. Try ItGiven (Figure), identify the domain and range using interval notation. Figure 12. Show Solution domain =[1950,2002] range = [47,000,000,89,000,000] Can a function’s domain and range be the same? Yes. For example, the domain and range of the cube root function are both the set of all real numbers. Finding Domains and Ranges of the Toolkit FunctionsWe will now return to our set of toolkit functions to determine the domain and range of each. Figure 13. For the constant function[latex]\,f\left(x\right)=c,\,[/latex]the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant[latex]\,c,\,[/latex]so the range is the set[latex]\,\left\{c\right\}\,[/latex]that contains this single element. In interval notation, this is written as[latex]\,\left[c,c\right],\,[/latex]the interval that both begins and ends with[latex]\,c.[/latex] Figure 14. For the identity function f(x)=x, there is no restriction on x. Both the domain and range are the set of all real numbers. Figure 15. For the absolute value function[latex]\,f\left(x\right)=|x|,\,[/latex]there is no restriction on[latex]\,x.\,[/latex]However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0. Figure 16. For the quadratic function[latex]\,f\left(x\right)={x}^{2},\,[/latex]the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers. Figure 17. For the cubic function[latex]\,f\left(x\right)={x}^{3},\,[/latex]the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers. Figure 18. For the reciprocal function[latex]\,f\left(x\right)=\frac{1}{x},\,[/latex]we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write[latex]\left\{x|\text{ }x\ne 0\right\},[/latex]the set of all real numbers that are not zero. Figure 19. For the reciprocal squared function[latex]\,f\left(x\right)=\frac{1}{{x}^{2}},[/latex]we cannot divide by [latex]0,[/latex] so we must exclude [latex]0[/latex] from the domain. There is also no [latex]x[/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers. Figure 20. For the square root function[latex]\,f\left(x\right)=\sqrt[]{x},\,[/latex]we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number[latex]\,x\,[/latex]is defined to be positive, even though the square of the negative number[latex]\,-\sqrt{x}\,[/latex]also gives us[latex]\,x.[/latex] Figure 21. For the cube root function[latex]\,f\left(x\right)=\sqrt[3]{x},\,[/latex]the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). How ToGiven the formula for a function, determine the domain and range.
Finding the Domain and Range Using Toolkit FunctionsFind the domain and range of[latex]\,f\left(x\right)=2{x}^{3}-x.[/latex] Show Solution There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result. The domain is[latex]\,\left(-\infty ,\infty \right)\,[/latex]and the range is also[latex]\,\left(-\infty ,\infty \right).[/latex] Finding the Domain and RangeFind the domain and range of[latex]\,f\left(x\right)=\frac{2}{x+1}.[/latex] Show Solution We cannot evaluate the function at[latex]\,-1\,[/latex]because division by zero is undefined. The domain is[latex]\,\left(-\infty ,-1\right)\cup \left(-1,\infty \right).\,[/latex]Because the function is never zero, we exclude 0 from the range. The range is[latex]\,\left(-\infty ,0\right)\cup \left(0,\infty \right).[/latex] Finding the Domain and RangeFind the domain and range of[latex]\,f\left(x\right)=2\sqrt{x+4}.[/latex] Show Solution We cannot take the square root of a negative number, so the value inside the radical must be nonnegative. [latex]x+4\ge 0\text{ when }x\ge -4[/latex] The domain of[latex]\,f\left(x\right)\,[/latex]is[latex]\,\left[-4,\infty \right).[/latex] We then find the range. We know that[latex]\,f\left(-4\right)=0,\,[/latex]and the function value increases as[latex]\,x\,[/latex]increases without any upper limit. We conclude that the range of[latex]\,f\,[/latex]is[latex]\,\left[0,\infty \right).[/latex] Analysis(Figure) represents the function[latex]\,f.[/latex] Figure 22. Try ItFind the domain and range of[latex]\,f\left(x\right)=-\sqrt{2-x}.[/latex] Show Solution domain:[latex]\,\left(-\infty ,2\right];\,[/latex]range:[latex]\,\left(-\infty ,0\right][/latex] Graphing Piecewise-Defined FunctionsSometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function[latex]\,f\left(x\right)=|x|.\,[/latex]With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0. If we input 0, or a positive value, the output is the same as the input. [latex]f\left(x\right)=x\,\text{if}\,x\ge 0[/latex] If we input a negative value, the output is the opposite of the input. [latex]f\left(x\right)=-x\,\text{if}\,x<0[/latex] Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain. We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income[latex]\,S\,[/latex]would be[latex]\,0.1S\,[/latex]if[latex]\,S\le \text{\$}10\text{,}000\,[/latex]and[latex]\,\text{\$}1000+0.2\left(S-\text{\$}10\text{,}000\right)\,[/latex]if[latex]\,S>\text{\$}10\text{,}000.[/latex] Piecewise FunctionA piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this: [latex]f\left(x\right)=\Bigg\{\begin{array}{l}\text{formula 1 if }x\text{ is in domain 1}\\ \text{formula 2 if }x\text{ is in domain 2}\\ \text{formula 3 if }x\text{ is in domain 3}\end{array}[/latex] In piecewise notation, the absolute value function is [latex]|x|=\bigg\{\begin{array}{l}x\text{ if }x\ge 0\\ -x\text{ if }x<0\end{array}[/latex] How ToGiven a piecewise function, write the formula and identify the domain for each interval.
Writing a Piecewise FunctionA museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people,[latex]\,n,\,[/latex]to the cost,[latex]\,C.[/latex] Show Solution Two different formulas will be needed. For n-values under 10,[latex]\,C=5n.\,[/latex]For values of[latex]\,n\,[/latex]that are 10 or greater,[latex]\,C=50.[/latex] [latex]C\left(n\right)=\left\{\begin{array}{ccc}5n& \text{if}& 0<n<10\\ 50& \text{if}& n\ge 10\end{array}[/latex] AnalysisThe function is represented in (Figure). The graph is a diagonal line from[latex]\,n=0\,[/latex]to[latex]\,n=10\,[/latex]and a constant after that. In this example, the two formulas agree at the meeting point where[latex]\,n=10,\,[/latex]but not all piecewise functions have this property. Figure 23. Working with a Piecewise FunctionA cell phone company uses the function below to determine the cost,[latex]\,C,\,[/latex]in dollars for[latex]\,g\,[/latex]gigabytes of data transfer. [latex]C\left(g\right)=\left\{\begin{array}{ccc}25& \text{if}& 0<g<2\\ 25+10\left(g-2\right)& \text{if}& g\ge 2\end{array}[/latex] Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data. Show Solution To find the cost of using 1.5 gigabytes of data,[latex]\,C\left(1.5\right),\,[/latex]we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula. [latex]C\left(1.5\right)=\text{\$}25[/latex] To find the cost of using 4 gigabytes of data,[latex]\,C\left(4\right),\,[/latex]we see that our input of 4 is greater than 2, so we use the second formula. [latex]C\left(4\right)=25+10\left(4-2\right)=\text{\$}45[/latex] AnalysisThe function is represented in (Figure). We can see where the function changes from a constant to a shifted and stretched identity at[latex]\,g=2.\,[/latex]We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain. Figure 24. How ToGiven a piecewise function, sketch a graph.
Graphing a Piecewise FunctionSketch a graph of the function. [latex]f\left(x\right)=\left\{\begin{array}{ccc}{x}^{2}& \text{if}& x\le 1\\ 3& \text{if}& 1<x\le 2\\ x& \text{if}& x>2\end{array}[/latex] Show Solution Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality. (Figure) shows the three components of the piecewise function graphed on separate coordinate systems. Figure 25. (a)[latex]\,f\left(x\right)={x}^{2}\text{ if }x\le 1;\,[/latex](b)[latex]\,f\left(x\right)=3\text{ if 1< }x\le 2;\,[/latex](c)[latex]\,f\left(x\right)=x\text{ if }x>2[/latex] Now that we have sketched each piece individually, we combine them in the same coordinate plane. See (Figure). Figure 26. AnalysisNote that the graph does pass the vertical line test even at[latex]\,x=1\,[/latex]and[latex]\,x=2\,[/latex]because the points [latex]\left(1,3\right)[/latex] and [latex]\left(2,2\right)[/latex] are not part of the graph of the function, though [latex]\left(1,1\right)[/latex] Try ItGraph the following piecewise function. [latex]f\left(x\right)=\left\{\begin{array}{ccc}{x}^{3}& \text{if}& x<-1\\ -2& \text{if}& -1<x<4\\ \sqrt{x}& \text{if}& x>4\end{array}[/latex] Show Solution Can more than one formula from a piecewise function be applied to a value in the domain? No. Each value corresponds to one equation in a piecewise formula. Access these online resources for additional instruction and practice with domain and range.
Key Concepts
Section ExercisesVerbalWhy does the domain differ for different functions? Show Solution The domain of a function depends upon what values of the independent variable make the function undefined or imaginary. How do we determine the domain of a function defined by an equation? Explain why the domain of[latex]\,f\left(x\right)=\sqrt[3]{x}\,[/latex]is different from the domain of[latex]\,f\left(x\right)=\sqrt[]{x}.[/latex] Show Solution There is no restriction on[latex]\,x\,[/latex]for[latex]\,f\left(x\right)=\sqrt[3]{x}\,[/latex]because you can take the cube root of any real number. So the domain is all real numbers,[latex]\,\left(-\infty ,\infty \right).\,[/latex]When dealing with the set of real numbers, you cannot take the square root of negative numbers. So[latex]\,x[/latex]-values are restricted for[latex]\,f\left(x\right)=\sqrt[]{x}\,[/latex]to nonnegative numbers and the domain is[latex]\,\left[0,\infty \right).[/latex] When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket? How do you graph a piecewise function? Show Solution Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the[latex]\,x[/latex]-axis and[latex]\,y[/latex]-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate[latex]\,-\infty \,[/latex]or[latex]\,\text{ }\infty .\,[/latex]Combine the graphs to find the graph of the piecewise function. AlgebraicFor the following exercises, find the domain of each function using interval notation. [latex]f\left(x\right)=-2x\left(x-1\right)\left(x-2\right)[/latex] [latex]f\left(x\right)=5-2{x}^{2}[/latex] Show Solution [latex]\left(-\infty ,\infty \right)[/latex] [latex]f\left(x\right)=3\sqrt{x-2}[/latex] [latex]f\left(x\right)=3-\sqrt{6-2x}[/latex] Show Solution [latex]\left(-\infty ,3\right][/latex] [latex]f\left(x\right)=\sqrt{4-3x}[/latex] [latex]\begin{array}{l}\\ f\left(x\right)=\sqrt[]{{x}^{2}+4}\end{array}[/latex] Show Solution [latex]\left(-\infty ,\infty \right)[/latex] [latex]f\left(x\right)=\sqrt[3]{1-2x}[/latex] [latex]f\left(x\right)=\sqrt[3]{x-1}[/latex] Show Solution [latex]\left(-\infty ,\infty \right)[/latex] [latex]f\left(x\right)=\frac{9}{x-6}[/latex] [latex]f\left(x\right)=\frac{3x+1}{4x+2}[/latex] Show Solution [latex]\left(-\infty ,-\frac{1}{2}\right)\cup \left(-\frac{1}{2},\infty \right)[/latex] [latex]f\left(x\right)=\frac{\sqrt{x+4}}{x-4}[/latex] [latex]f\left(x\right)=\frac{x-3}{{x}^{2}+9x-22}[/latex] Show Solution [latex]\left(-\infty ,-11\right)\cup \left(-11,2\right)\cup \left(2,\infty \right)[/latex] [latex]f\left(x\right)=\frac{1}{{x}^{2}-x-6}[/latex] [latex]f\left(x\right)=\frac{2{x}^{3}-250}{{x}^{2}-2x-15}[/latex] Show Solution [latex]\left(-\infty ,-3\right)\cup \left(-3,5\right)\cup \left(5,\infty \right)[/latex] [latex]\frac{5}{\sqrt{x-3}}[/latex] [latex]\frac{2x+1}{\sqrt{5-x}}[/latex] Show Solution [latex]\left(-\infty ,5\right)[/latex] [latex]f\left(x\right)=\frac{\sqrt{x-4}}{\sqrt{x-6}}[/latex] [latex]f\left(x\right)=\frac{\sqrt{x-6}}{\sqrt{x-4}}[/latex] Show Solution [latex]\left[6,\infty \right)[/latex] [latex]f\left(x\right)=\frac{x}{x}[/latex] [latex]f\left(x\right)=\frac{{x}^{2}-9x}{{x}^{2}-81}[/latex] Show Solution [latex]\left(-\infty ,-9\right)\cup \left(-9,9\right)\cup \left(9,\infty \right)[/latex] Find the domain of the function[latex]\,f\left(x\right)=\sqrt{2{x}^{3}-50x}\,[/latex]by:
GraphicalFor the following exercises, write the domain and range of each function using interval notation. Show Solution domain:[latex]\,\left(2,8\right],\,[/latex]range[latex]\,\left[6,8\right)\,[/latex] Show Solution domain:[latex]\,\left[-4,\text{ 4],}\,[/latex]range:[latex]\,\left[0,\text{ 2]}[/latex] Show Solution domain:[latex]\,\left[-5,\text{ }3\right),\,[/latex]range:[latex]\,\left[0,2\right][/latex] Show Solution domain:[latex]\,\left(-\infty ,1\right],\,[/latex]range:[latex]\,\left[0,\infty \right)\,[/latex] Show Solution domain:[latex]\,\left[-6,-\frac{1}{6}\right]\cup \left[\frac{1}{6},6\right];\,[/latex]range:[latex]\,\left[-6,-\frac{1}{6}\right]\cup \left[\frac{1}{6},6\right]\,[/latex] Show Solution domain:[latex]\,\left[-3,\text{ }\infty \right);\,[/latex]range:[latex]\,\left[0,\infty \right)\,[/latex] For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation. [latex]f\left(x\right)=\left\{\begin{array}{lll}x+1\hfill & \text{if}\hfill & x<-2\hfill \\ -2x-3\hfill & \text{if}\hfill & x\ge -2\hfill \end{array}[/latex] [latex]f\left(x\right)=\left\{\begin{array}{lll}2x-1\hfill & \text{if}\hfill & x<1\hfill \\ 1+x\hfill & \text{if}\hfill & x\ge 1\hfill \end{array}[/latex] Show Solution domain:[latex]\,\left(-\infty ,\infty \right)[/latex] [latex]f\left(x\right)=\left\{\begin{array}{c}x+1\,\,\text{if}\,\,x<0\\ x-1\,\,\text{if}\,\,\,x>0\end{array}[/latex] [latex]f\left(x\right)=\left\{\begin{array}{ccc}3& \text{if}& x<0\\ \sqrt{x}& \text{if}& x\ge 0\end{array}[/latex] Show Solution domain:[latex]\,\left(-\infty ,\infty \right)[/latex] [latex]f\left(x\right)=\left\{\begin{array}{c}{x}^{2}\text{ if }x<0\\ 1-x\text{ if }x>0\end{array}[/latex] [latex]f\left(x\right)=\left\{\begin{array}{r}\hfill \begin{array}{r}\hfill {x}^{2}\\ \hfill x+2\end{array}\end{array}\,\,\begin{array}{l}\text{if}\,\,\,\,\,x<0\hfill \\ \text{if}\,\,\,\,\,x\ge 0\hfill \end{array}[/latex] Show Solution domain:[latex]\,\left(-\infty ,\infty \right)[/latex] [latex]f\left(x\right)=\left\{\begin{array}{ccc}x+1& \text{if}& x<1\\ {x}^{3}& \text{if}& x\ge 1\end{array}[/latex] [latex]f\left(x\right)=\left\{\begin{array}{c}|x|\\ 1\end{array}\begin{array}{l}\,\,\,\text{if}\,\,\,x<2\hfill \\ \,\,\,\text{if}\,\,\,x\ge 2\hfill \end{array}[/latex] Show Solution domain:[latex]\,\left(-\infty ,\infty \right)[/latex] NumericFor the following exercises, given each function [latex]f,[/latex]evaluate [latex]f\left(-3\right),\,f\left(-2\right),\,f\left(-1\right),[/latex] and [latex]f\left(0\right).[/latex] [latex]f\left(x\right)=\left\{\begin{array}{lll}x+1\hfill & \text{if}\hfill & x<-2\hfill \\ -2x-3\hfill & \text{if}\hfill & x\ge -2\hfill \end{array}[/latex] [latex]f\left(x\right)=\left\{\begin{array}{cc}1& \text{if }x\le -3\\ 0& \text{if }x>-3\end{array}[/latex] Show Solution [latex]\begin{array}{cccc}f\left(-3\right)=1;& f\left(-2\right)=0;& f\left(-1\right)=0;& f\left(0\right)=0\end{array}[/latex] [latex]f\left(x\right)=\left\{\begin{array}{cc}-2{x}^{2}+3& \text{if }x\le -1\\ 5x-7& \text{if }x>-1\end{array}[/latex] For the following exercises, given each function[latex]\,f,\,[/latex]evaluate[latex]f\left(-1\right),\,f\left(0\right),\,f\left(2\right),\,[/latex]and[latex]\,f\left(4\right).[/latex] [latex]f\left(x\right)=\left\{\begin{array}{lll}7x+3\hfill & \text{if}\hfill & x<0\hfill \\ 7x+6\hfill & \text{if}\hfill & x\ge 0\hfill \end{array}[/latex] Show Solution [latex]\begin{array}{cccc}f\left(-1\right)=-4;& f\left(0\right)=6;& f\left(2\right)=20;& f\left(4\right)=34\end{array}[/latex] [latex]f\left(x\right)=\left\{\begin{array}{ccc}{x}^{2}-2& \text{if}& x<2\\ 4+|x-5|& \text{if}& x\ge 2\end{array}[/latex] [latex]f\left(x\right)=\left\{\begin{array}{ccc}5x& \text{if}& x<0\\ 3& \text{if}& 0\le x\le 3\\ {x}^{2}& \text{if}& x>3\end{array}[/latex] Show Solution [latex]\begin{array}{cccc}f\left(-1\right)=-5;& f\left(0\right)=3;& f\left(2\right)=3;& f\left(4\right)=16\end{array}[/latex] For the following exercises, write the domain for the piecewise function in interval notation. [latex]f\left(x\right)=\left\{\begin{array}{c}x+1\,\,\,\,\,\text{ if}\,\,x<-2\\ -2x-3\,\,\text{if}\,\,x\ge -2\end{array}[/latex] [latex]f\left(x\right)=\left\{\begin{array}{c}{x}^{2}-2\,\,\,\,\,\text{ if}\,\,x<1\\ -{x}^{2}+2\,\,\text{if}\,\,x>1\end{array}[/latex] Show Solution domain:[latex]\,\left(-\infty ,1\right)\cup \left(1,\infty \right)[/latex] [latex]f\left(x\right)=\left\{\begin{array}{c}2x-3\\ -3{x}^{2}\end{array}\,\,\begin{array}{c}\text{if}\,\,\,x<0\\ \text{if}\,\,\,x\ge 2\end{array}[/latex] TechnologyGraph[latex]\,y=\frac{1}{{x}^{2}}\,[/latex]on the viewing window[latex]\,\left[-0.5,-0.1\right]\,[/latex]and[latex]\,\left[0.1,0.5\right].\,[/latex]Determine the corresponding range for the viewing window. Show the graphs. Show Solution window: [−0.5,−0.1]; [−0.5,−0.1]; [−0.5,−0.1]; range: [4, 100] [4, 100] [4, 100] window: [0.1, 0.5]; [0.1, 0.5]; [0.1, 0.5]; range: [4, 100] [4, 100] [4, 100] Graph[latex]\,y=\frac{1}{x}\,[/latex]on the viewing window[latex]\,\left[-0.5,-0.1\right]\,[/latex]and[latex]\,\left[0.1,\text{ }0.5\right].\,[/latex]Determine the corresponding range for the viewing window. Show the graphs. ExtensionSuppose the range of a function[latex]\,f\,[/latex]is[latex]\,\left[-5,\text{ }8\right].\,[/latex]What is the range of[latex]\,|f\left(x\right)|?[/latex] Show Solution [latex]\left[0,\text{ }8\right][/latex] Create a function in which the range is all nonnegative real numbers. Create a function in which the domain is[latex]\,x>2.[/latex] Show Solution Many answers. One function is[latex]\,f\left(x\right)=\frac{1}{\sqrt{x-2}}.[/latex] Real-World ApplicationsThe height[latex]\,h\,[/latex]of a projectile is a function of the time[latex]\,t\,[/latex]it is in the air. The height in feet for[latex]\,t\,[/latex]seconds is given by the function[latex]h\left(t\right)=-16{t}^{2}+96t.[/latex] Show Solution The domain is[latex]\,\left[0,\text{ }6\right];\,[/latex]it takes 6 seconds for the projectile to leave the ground and return to the ground The cost in dollars of making[latex]\,x\,[/latex]items is given by the function[latex]\,C\left(x\right)=10x+500.[/latex] What is the domain of the function described?The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of values that the function assumes.
What is the domain of a rational function the range?The domain of a rational function is the set of inputs (x-values) that do not cause a zero denominator. For example, the domain of f(x) = 1/x is all real numbers except x = 0, often denoted as R – {0}. The range of a rational function is the set of all possible outputs (y-values).
What is the domain of a rational function quizlet?What is the domain of a rational function? All real numbers except those for which the denominator q is zero.
How do you find the domain of a rational function example?The domain of a rational function is the set of all x-values that the function can take. To find the domain of a rational function y = f(x): Set the denominator ≠ 0 and solve it for x. Set of all real numbers other than the values of x mentioned in the last step is the domain.
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