CHAPTER 5 SECTION 5.3 INVERSE FUNCTIONS

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CHAPTER 5SECTION 5.3INVERSE FUNCTIONS
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Definition of Inverse Function and Figure 5.10
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1. Show that these functions are inverses
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1. Show that these functions are inverses

• You could
• Graph each and show symmetry about y x.
• Show that both f(g(x)) x and g(f(x)) x.
• Find the inverse of one of the functions and
  compare.

• To find an inverse
• Swap x y.
• Solve for y.
• Domain of f -1(x) Range of f(x).

These functions are inverses.
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Theorem 5.6 Reflective Property of Inverse
Functions
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Theorem 5.7 The Existence of an Inverse Function
and Figure 5.13
Do you remember what monotonic means!!!!!!!!!!????
??????????
PASSES HOROZONTAL LINE TEST!!
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Inverse Functions

• If f(g(x)) x and g(f(x)) x then f(x) and g(x)
  are inverses.
• Domain of f(x) Range of f -1(x)
• Range of f(x) Domain of f -1(x)
• Inverses are symmetric about y x.
• A function can only have an inverse if it is
  1-to-1.
• 2 ways to check 1-to-1
• a.
• b.

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Inverse Functions

• If f(g(x)) x and g(f(x)) x then f(x) and g(x)
  are inverses.
• Domain of f(x) Range of f -1(x)
• Range of f(x) Domain of f -1(x)
• Inverses are symmetric about y x.
• A function can only have an inverse if it is
  1-to-1.
• 2 ways to check 1-to-1
• a. horizontal line test
• b. is it always inc or dec?
• Note if a function isnt 1-to-1 we can change
  its domain to make it 1-to-1.

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Can these functions have inverses?
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Can these functions have inverses?
This function cant have an inverse.
This derivative is always positive, so y is
always increasing.
This derivative changes signs, so y increases and
decreases.
This function can have an inverse.
This function cant have an inverse.
unless we limit its domain to all positives or
all negatives.
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Guidelines for Finding an Inverse Function
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5. Find the inverse of
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5. Find the inverse of
This function isnt 1-to-1.
Limit its domain.
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Theorem 5.8 Continuity and Differentiability of
Inverse Functions
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Theorem 5.9 The Derivative of an Inverse Function
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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 Th 5.9,
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)21) 1/4
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We showed previously that these functions are
inverses.
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We showed previously that these functions are
inverses.
Conclusion
If (a,b) is on f(x), that means (b,a) is on f
-1(x), AND
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• To find
• a. Set b f(a) .
• Solve for a.

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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
Graphically f and its inverse should look like
mirror images across the line y x.
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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
Graphically f and its inverse should look like
mirror images across the line y x.
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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
Graphically f and its inverse should look like
mirror images across the line y x.
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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
Graphically f and its inverse should look like
mirror images across the line y x.
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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
Graphically f and its inverse should look like
mirror images across the line y x.
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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
Graphically f and its inverse should look like
mirror images across the line y x.
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55. 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 ).
On (0, )
The derivative is always negative on (0, ),
therefore, f is decreasing and thus has an
inverse on this interval.
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55. 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 ).
On (0, )
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55. 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 ).
On (0, )
The derivative is always negative on (0, ),
therefore, f is decreasing and thus has an
inverse on this interval.
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Some function f(x) and its
derivative f '(x) are continuous and
differentiable for all real numbers, and some of
the values for the functions are given in the
below table Based on the information
given, answer the following questions(a)
Evaluate (b) At what value c is the graph of
discontinuous?You may not use a
graphing calculator Difficulty
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• Remember that                        , allowing
  us to find the derivative of an inverse function
  given only the original function. However, you'll
  need to be able to compute     .
• Remember, a function and its inverse differ in
  that the input for one is the output for the
  other. Since the above table tells us that f(1)
  3, we can be sure that
• Plug what you know into the formula we began part
  (a) with and we get
•                                    
• The table tells us that f '(1) 2, so     .
              .

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(b) The formula we used in part (a), written more
generally, says that
Therefore, whenever the denominator
equals 0, the fraction will be undefined. So, the
answer to our question is the equivalent to the
solution of the equation
           
To solve this, first decide at what value of x
does f ' equal 0. The only answer we can be sure
of is when x 2. Since f '(2) 0, we can then
say, by substitution, that            . This
means the exact same thing as f(2) a.
Basically, when it's all said and done, we're
just looking for f(2), so the answer is 1. To
check, try and evaluate             
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Obviously, that derivative doesn't exist.