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). .
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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
g(f(x)) = x for each x in the domain of f.
The function g is denoted by f-1 (read f inverse).
are inverses of each other. See how they mirror
each other across y = x.
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).
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.
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))
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
corresponds to an x value of 2, thus if (2,3) is on the
f function, (3,2) is on the function.
So, f-1 (3) = 2
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
Two inverse functions are:
Pick a point that satisfies
f, such as (3,9), then
(9,3) satisfies g.
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
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.