Answer Before 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:
Let f and g be two functions. If
f(g(x)) = x and g(f(x)) = x,
then g is the inverse of f and f is the inverse of g.
Exercise 1:
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Finding Inverses
Example 1. First consider a simple example f(x) = 3x + 2.
The graph of f is a line with slope 3, so it passes the horizontal line test and does have an inverse.
There are two steps required to evaluate f at a number x. First we multiply x by 3, then we add 2.
Thinking of the inverse function as undoing what f did, we must undo these steps in reverse order.
The steps required to evaluate f-1 are to first undo the adding of 2 by subtracting 2. Then we undo multiplication by 3 by dividing by 3.
Therefore, f-1(x) = (x - 2)/3.
Steps for finding the inverse of a function f.
Replace f(x) by y in the equation describing the function.
Interchange x and y. In other words, replace every x by a y and vice versa.
Solve for y.
Replace y by f-1(x).
Example 2. f(x) = 6 - x/2
Step 1 y = 6 - x/2.
Step 2 x = 6 - y/2.
Step 3 x = 6 - y/2.
y/2 = 6 - x.
y = 12 - 2x.
Step 4 f-1(x) = 12 - 2x.
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
This is the function we worked with in Exercise 1. From its graph (shown above) we see that it does have an inverse. (In fact, its inverse was given in Exercise 1.)
Step 1 y = x3 + 2.
Step 2 x = y3 + 2.
Step 3 x - 2 = y3.
(x - 2)^(1/3) = y.
Step 4 f-1(x) = (x - 2)^(1/3).
Exercise 3:
Graph f(x) = 1 - 2x3 to see that it does have an inverse. Find f-1(x). Answer
Step-by-step explanation:
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