Inverse Functions

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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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Graphically speaking the yellow and red
graphs are inverses of each other. See how they
mirror each other across y x.
y1/(x-2)
y x
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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).
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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))
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Example
What is the value of f-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.
Lets look at the graph.
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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)
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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
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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.
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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
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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