Inverse Functions

1. Inverse Functions Definition of Inverse Functions A function g is the inverse of the function f if f(g(x)) = x for each x in the domain of g and g(f(x)) = x for each x in the domain of f. The function g is denoted by f-1 (read f inverse).

2. Graphically speaking the yellow and red graphs are inverses of each other. See how they mirror each other across y = x. y=1/(x-2) y= x

3. Theorem Continuity and Differentiability of Inverse Functions Let f be a function whose domain is an interval I. If f has an inverse, them the following statements are true 1. If f is continuous on its domain, then f-1 is continuous on its domain. 2. If f is increasing on its domain, then f-1 is increasing on its domain. 3. If f is decreasing on its domain, then f-1 is decreasing on its domain. 4. If f is differentiable at c and f-1 (c) does not equal 0 then f-1 is differentiable at f(c).

4. Theorem The Derivative of an Inverse Function Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which f’ (g(x)) is not zero. Moreover, g’ (x) = 1/ f’ (g(x)). Proof: Since g is the inverse of f, f(g(x)) = x Taking the derivative of both sides with respect to x, we get f’ (g(x))g’ (x) = 1 Thus g’ (x) = 1/ f’ (g(x))

5. Example What is the value off-1 (x) when x = 3? Since we want the inverse, 3 would be the y coordinate of some value of x in f(x). As you can see, we could try to guess an answer but we have no means to solve the equation. Let’s look at the graph.

6. On the graph you can see that a y value of 3 corresponds to an x value of 2, thus if (2,3) is on the f function, (3,2) is on the function. f-1 So, f-1 (3) = 2 (2,3)

7. B. What is the value of (f-1)’ (x) when x = 3? Solution: Since g’ (x) = 1/ f’ (g(x)) by our previous theorems, we can substitute f-1 for g, thus [f-1 (x)]’ = 1/ f’ (f-1 (x)) [f-1 (3)]’ = 1/ f’ (f-1 (3)) = 1/ f’(2) = 1/(3/4(2)(2)+1) = 1/4

8. Graphs of Inverse Functions Have Reciprocal Slopes Two inverse functions are: Pick a point that satisfies f, such as (3,9), then (9,3) satisfies g.

9. Homework Examples: 4. Show that f and g are inverse functions (a) algebraically and (b) graphically Solution: One way to do (a) is to show that f(g(x))=x and g(f(x)) = x. A second method would be to find the inverse of f and show that it is g. Four steps to finding an Inverse: Step 1 change f(x) to y Step 2 Interchange x and y Step 3 solve for y Step 4 change y to f-1

10. Graphically: f and its inverse should look like mirror images across the line y = x. Show that f is strictly monotonic on the indicated interval and therefore has an inverse on that interval. (Strictly monotonic means that f is always increasing on a given interval or f is always decreasing on a given interval ). The derivative is always negative on , therefore, f is decreasing and thus has an inverse on this interval. On