Contents: This page corresponds to § 1.7 (p. 150) of the text. Show
Suggested Problems from Text p.158 #1-4, 5, 8, 9, 12, 13, 15, 18, 21, 22, 27, 31, 34, 37, 46, 48, 51, 71, 74, 83
Definition of Inverse FunctionBefore defining the inverse of a function we need to have the right mental image of function. Consider the function f(x) = 2x + 1. We know how to evaluate f at 3, f(3) = 2*3 + 1 = 7. In this section it helps to think of f as transforming a 3 into a 7, and f transforms a 5 into an 11, etc. Now that we think of f as "acting on" numbers and transforming them, we can define the inverse of f as the function that "undoes" what f did. In other words, the inverse of f needs to take 7 back to 3, and take -3 back to -2, etc. Let g(x) = (x - 1)/2. Then g(7) = 3, g(-3) = -2, and g(11) = 5, so g seems to be undoing what f did, at least for these three values. To prove that g is the inverse of f we must show that this is true for any value of x in the domain of f. In other words, g must take f(x) back to x for all values of x in the domain of f. So, g(f(x)) = x must hold for all x in the domain of f. The way to check this condition is to see that the formula for g(f(x)) simplifies to x. g(f(x)) = g(2x + 1) = (2x + 1 -1)/2 = 2x/2 = x. This simplification shows that if we choose any number and let f act it, then applying g to the result recovers our original number. We also need to see that this process works in reverse, or that f also undoes what g does. f(g(x)) = f((x - 1)/2) = 2(x - 1)/2 + 1 = x - 1 + 1 = x. Letting f-1 denote the inverse of f, we have just shown that g = f-1. Definition:
Exercise 1:
Return to Contents Graphs of Inverse FunctionsWe have seen examples of reflections in the plane. The reflection of a point (a,b) about the x-axis is (a,-b), and the reflection of (a,b) about the y-axis is (-a,b). Now we want to reflect about the line y = x.
Let f(x) = x3 + 2. Then f(2) = 10 and the point (2,10) is on the graph of f. The inverse of f must take 10 back to 2, i.e. f-1(10)=2, so the point (10,2) is on the graph of f-1. The point (10,2) is the reflection in the line y = x of the point (2,10). The same argument can be made for all points on the graphs of f and f-1. The graph of f-1 is the reflection about the line y = x of the graph of f.
Return to Contents Existence of an InverseSome functions do not have inverse functions. For example, consider f(x) = x2. There are two numbers that f takes to 4, f(2) = 4 and f(-2) = 4. If f had an inverse, then the fact that f(2) = 4 would imply that the inverse of f takes 4 back to 2. On the other hand, since f(-2) = 4, the inverse of f would have to take 4 to -2. Therefore, there is no function that is the inverse of f. Look at the same problem in terms of graphs. If f had an inverse, then its graph would be the reflection of the graph of f about the line y = x. The graph of f and its reflection about y = x are drawn below. Note that the reflected graph does not pass the vertical line test, so it is not the graph of a function. This generalizes as follows: A function f has an inverse if and only if when its graph is reflected about the line y = x, the result is the graph of a function (passes the vertical line test). But this can be simplified. We can tell before we reflect the graph whether or not any vertical line will intersect more than once by looking at how horizontal lines intersect the original graph! Horizontal Line Test
The property of having an inverse is very important in mathematics, and it has a name. Definition: A function f is one-to-one if and only if f has an inverse. The following definition is equivalent, and it is the one most commonly given for one-to-one. Alternate Definition: A function f is one-to-one if, for every a and b in its domain, f(a) = f(b) implies a = b.
Return to Contents Finding InversesExample 1. First consider a simple example f(x) = 3x + 2.
Steps for finding the inverse of a function f.
Step 2 often confuses students. We could omit step 2, and solve for x instead of y, but then we would end up with a formula in y instead of x. The formula would be the same, but the variable would be different. To avoid this we simply interchange the roles of x and y before we solve. Example 3. f(x) = x3 + 2
Exercise 3:
Return to Contents How do you know if the inverse of something is a function?If there is any place on the graph of the original function where a horizontal line would cross two more more times, then the inverse of that function will not itself be a function. If all horizontal lines cross at most one spot on the original graph, then the inverse will be a function, too.
What is the inverse of the table?For a table, the x-values of the function are the y-values of its inverse, and the y-values of the function are the x-values of its inverse.
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