Expert Answer:Composite, One-to-One and Inverse Functions

  

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Composite, One-to-one & Inverse Functions
Section 5.1: Composite Functions
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Form a Composite Function (14:52)
Post 1
Exploration 1*: Form a Composite Function
1. Suppose you have a job that pays $10 per hour. Write a function, g that can be used to determine
your gross pay (your pay before taxes are taken out) per hour, h, that you worked.
g (h) =
2. Now let’s write a formula for how much money you’ll actually take home of that paycheck.
Let’s assume your employer withholds 20% of your gross pay for taxes. Write a function, n, that
determines your net pay based off of your gross income, g.
n( g ) =
3. How much money would you net if you worked for 20 hours?
4. Instead of having to use two different functions to find out your net pay, as you most likely did in
(3), let’s combine our functions from (1) and (2) and write them as one function. This is called
composing functions.
Write a function that relates the number of hours worked, h, to your net pay, n.
Copyright © 2016 Pearson Education, Inc.
Section 5.1 & 5.2
Definition: Given two functions f and g, the composite function, denoted by __________ (read as “f
composed with g”) is defined by________________________.
Note: 𝑓 ∘ 𝑔 does not mean f multiplied by g(x). It means input the function g into the function f:
𝑓 ∘ 𝑔 = 𝑓(𝑔(𝑥)) ≠ 𝑓(𝑥)𝑔(𝑥)
The domain of 𝑓 ∘ 𝑔 is the set of all numbers x in the domain of g such that g ( x ) is in the domain of f.
In other words, 𝑓 ∘ 𝑔 is defined whenever both g ( x ) and f ( g ( x ) ) are defined.
Example 1*: Form a Composite Function; Evaluate a Composite Function
Suppose that f ( x ) = 2 x 2 + 3 and g ( x ) = 4 x3 + 1 . Find:
(a) (𝑓 ∘ 𝑔)(1)
(b) (𝑔 ∘ 𝑓)(1)
(c) (𝑓 ∘ 𝑓)(−2)
(d) (𝑔 ∘ 𝑔)(−1)
End of Post 1
Example 2: Evaluate a Composite Function
If f (x) and g(x) are polynomial functions, use the table of values for f (x) and g(x) to complete the table
of values for (𝑓 ∘ 𝑔)(𝑥).
x
g ( x)
x
-2
-1
0
1
2
4
1
0
1
4
0
1
2
3
4
f ( x)
3
4
5
6
7
x
-2
-1
0
1
3
Copyright © 2016 Pearson Education, Inc.
(𝑓 ∘ 𝑔)(𝑥)
Composite, One-to-one & Inverse Functions
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Find the Domain of a Composite Function (11:50)
Post 2
Example 3: Find the Domain of a Composite Function
Suppose that f ( x ) = 2 x 2 + 3 and g ( x ) = 4 x3 + 1 . Find the following and their domains:
(a) 𝑓 ∘ 𝑔
(b) 𝑔 ∘ 𝑓
Example 4*: Find the Domain of a Composite Function
1
4
and g ( x ) =
Find the domain of (𝑓 ∘ 𝑔)(𝑥) if f ( x ) =
.
x+4
x−2
End of Post 2
Example 5: Find a Composite Function and Its Domain
1
Suppose that f ( x ) = and g ( x ) = x − 1 . Find the following and their domains:
x
(a) 𝑓 ∘ 𝑔
(b) 𝑓 ∘ 𝑓
Copyright © 2016 Pearson Education, Inc.
Section 5.1 & 5.2
Know these examples.
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Showing Two Composite Functions Are Equal (1:56)
Example 6*: Showing Two Composite Functions Are Equal
1
If f ( x ) = 2 x and g ( x ) = x, show that (𝑓 ∘ 𝑔)(𝑥) = (𝑔 ∘ 𝑓)(𝑥) = 𝑥 for every x in the domain of 𝑓 ∘ 𝑔
2
and 𝑔 ∘ 𝑓.
Example 7: Showing Two Composite Functions Are Equal
1
If f ( x ) = ( x − 1) and g ( x ) = 2 x + 1, show that (𝑓 ∘ 𝑔)(𝑥) = (𝑔 ∘ 𝑓)(𝑥) = 𝑥 for every x in the domain
2
of 𝑓 ∘ 𝑔 and 𝑔 ∘ 𝑓.
Copyright © 2016 Pearson Education, Inc.
Composite, One-to-one & Inverse Functions
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Find the Components of a Composite Function (1:27)
Example 8*: Find the Components of a Composite Function
Find functions f and g such that 𝑓 ∘ 𝑔 = 𝐻 if 𝐻(𝑥) = (2𝑥 + 3)4 .
Example 9: Find the Components of a Composite Function
1
Find functions f and g such that 𝑓 ∘ 𝑔 = 𝐻 if 𝐻(𝑥) = 2𝑥 2 −3.
Copyright © 2016 Pearson Education, Inc.
Section 5.1 & 5.2
Section 5.2: One-to-One Functions; Inverse Functions
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Determine Whether a Function Is One-to-One (12:53)
Post 3
Definition: A function is one – to – one if any two different inputs in the domain correspond to
_________________________________________________. That is, if x1 and x2 are two different
inputs of a function f, is one – to – one if __________________.
Example 1*: Determine Whether a Function is One – to – One
Determine whether the following functions are one – to – one. Explain why or why not.
(a)
Student
Car
Dan
Saturn
John
Pontiac
Joe
Honda
Andy
(b)
{(1,5), (2,8), (3,11), (4,14)}
The Horizontal Line Test Theorem: If every horizontal line intersects the graph of a function f in at
most ________________, then f is one – to – one.
End of Post 3
Copyright © 2016 Pearson Education, Inc.
Composite, One-to-one & Inverse Functions
Why does this test work? You may want to refer to the definition of one – to – one functions.
Example 2: Determine whether a Function is One – to – One Using the Horizontal Line Test
For each function, use the graph to determine whether the function is one – to – one.
Theorem: A function that is increasing on an interval I is a one – to – one function on I.
A function that is decreasing on an interval I is a one – to – one function on I.
Why is this theorem true?
Exploration 1: Inverse Functions – Reverse the Process
You might have experienced converting between degrees Fahrenheit and degrees Celsius when measuring
a temperature. The standard formula for determining temperature in degrees Fahrenheit, when given the
9
temperature in degrees Celsius, is F = C + 32 . We can use this formula to define a function named g,
5
9
namely F = g ( C ) = C + 32 , where C is the number of degrees Celsius and g ( C ) is a number of degrees
5
Fahrenheit. The function g defines a process for converting degrees Celsius to degrees Fahrenheit.
1. What is the value of g (100 ) ? What does it represent?
2. Solve the equation g ( C ) = 112 and describe the meaning of your answer.
3. What happens if you want to input degree Fahrenheit and output degree Celsius? Reverse the
9
process of the formula F = C + 32 by solving for C.
5
Copyright © 2016 Pearson Education, Inc.
Section 5.1 & 5.2
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Determine the Inverse of a Function (15:23)
Post 4
Definition: Suppose that f is a one – to – one function. Then, to each x in the domain of f, there is
_______________________ y in the range (because f is a function); and to each y in the range of f there
is exactly one x in the domain (because f is one – to – one). The correspondence from the range of f
back to the ______________ of f is called the inverse function of f. We use the symbol f −1 to denote
1
the inverse of f. Note: f −1 
f
In other words, two functions are said to be inverses of each other if they are the reverse process of
each other. Notice in the exploration, the formula found in part (c) was the reverse process of g.
Instead of inputting Celsius and outputting Fahrenheit, the new function inputs Fahrenheit and outputs
Celsius.
Example 3*: Determine the Inverse of a Function
Find the inverse of the following function. Let the domain of the function represent certain students, and
let the range represent the make of that student’s car. State the domain and the range of the inverse
function.
Student
Car
Dan
Saturn
John
Pontiac
Joe
Honda
Michelle
Chrysler
Example 4*: Determine the Inverse of a Function
Find the inverse of the following one – to – one function. Then state the domain and range of the
function and its inverse.
{(1,5), (2,8), (3,11), (4,14)}
Copyright © 2016 Pearson Education, Inc.
Composite, One-to-one & Inverse Functions
Domain and Range of Inverse Functions: Since the inverse function, f −1 , is a reverse mapping of the
function f :
Domain of f = _____________ of f −1
and
Range of f = _______________ of f −1
Fact: What f does, f −1 undoes and vice versa. Therefore,
f −1 ( f ( x ) ) = ____ where x is in the domain of f
f ( f −1 ( x ) ) = ____ where x is in the domain of f −1
We can use this fact to verify if two functions are inverses of each other.
Example 5*: Determine the Inverse of a Function ; Verifying Inverse Functions
Verify that the inverse of g ( x ) = x3 + 2 is g −1 ( x ) = 3 x − 2 by showing that g ( g −1 ( x ) ) = x for all x in
the domain of g and that g −1 ( g ( x ) ) = x for all x in the domain of g -1.
End of Post 4
Copyright © 2016 Pearson Education, Inc.
Section 5.1 & 5.2
Exploration 2: Graphs of Inverse Functions
1. Using a graphing utility, graph the following functions on the same screen
y = x, y = x3 , and y = 3 x
2. What do you notice about the graphs of y = x3 , its inverse y = 3 x , and the line y = x ?
3. Repeat this experiment by graphing the following functions on the same screen:
1
y = x, y = 2 x + 3, and y = ( x − 3)
2
1
4. What do you notice about the graphs of y = 2 x + 3, its inverse y = ( x − 3), and the line y = x ?
2
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Obtain the Graph of the Inverse Function (05:05)
Theorem: The graph of a one – to – one function f and the graph of its inverse f −1 are symmetric
with respect to the line ______________.
Example 6*: Obtain the Graph of the Inverse Function
The graph shown is that of a one – to – one function. Draw the graph of its inverse.
Copyright © 2016 Pearson Education, Inc.
Composite, One-to-one & Inverse Functions
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Find the Inverse Function from an Equation I (4:43)
Know this procedure.
Procedure for Finding the Inverse of a One – to – One Function
Step 1: In y = f ( x) , interchange the variables x and y to obtain ____________. This equation defines
the inverse function f −1 implicitly.
Step 2: If possible, solve the implicit equation for y in terms of x to obtain the explicit form of f −1 :
____________________.
Step 3: Check the result by showing that _______________ and ________________.
Example 7*: Find the Inverse Function from an Equation I
Find the inverse of f ( x) = 4 x + 2.
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Find the Inverse Function from an Equation II (9:10)
Post 5
Example 8*: Find the Inverse Function from an Equation II
2x −1
, x  -1 is one – to – one. Find its inverse and state the domain and range
The function f ( x) =
x +1
of both f and its inverse function.
End of Post 5
Copyright © 2016 Pearson Education, Inc.

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