Inverse FunctionsGraph Square/Cube Root Functions

Objectives:

1. To find the inverse of a function

2. To graph inverse functions

3. To graph square and cube root functions as transformations on parent functions

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Function Composition

Function composition happens when we take a whole function and substitute it in for x in another function.

– The “interior” function gets substituted in for x in the “exterior” function

Substitute in for in

Exercise 1

Let f(x) = 4x + 2 and g(x) = 1/4x – 1/2. Find the following compositions.

1. f(g(x)) 2. g(f(x))

Inverse Relations

An inverse relation is a relation that switches the inputs and output of another relation.

Inverse relations “undo” each other

Inverse Functions

If a relation and its inverse are both functions, then they are called inverse functions.

𝑓 ( 𝑓 − 1 (𝑥 ) )=𝑥∧ 𝑓 −1 ( 𝑓 (𝑥 ) )=𝑥

f -1 = “f inverse” or “inverse of f ”

The inverse of a function is not necessarily a

function.

Inverse Functions

If a relation and its inverse are both functions, then they are called inverse functions.

𝑓 ( 𝑓 − 1 (𝑥 ) )=𝑥∧ 𝑓 −1 ( 𝑓 (𝑥 ) )=𝑥

For and to be inverse functions, the domain of

must be equal to the range of , and the range of must be equal to the domain of .

Exercise 2

Verify that f(x) = 2x + 3 and f -1(x) = ½x – 3/2 are inverse functions.

Step 1

Step 2Step 1

Finding the Inverse of a FunctionSince the inverse of

a function switches the - and -values of the original function, we can easily find the inverse of a function algebraically:

Let , if necessary

Exchange the and variables

Solve for

Exercise 3

Find the inverse of f(x) = −(2/3) x + 2.

Exercise 4

Let . Find the inverse of . What is the relationship between the slopes of inverse linear functions?

Exercise 5

Find the inverse of the given function.

1. 2. 3.

Inverses of Nonlinear Functions

For some nonlinear functions, you have to first restrict the domain to find its inverse.

𝑦=𝑥2

𝑥=𝑦 2

±√𝑥=𝑦𝑦=√𝑥   or   𝑦=−√𝑥

?

Domain:

Inverses of Nonlinear Functions

For some nonlinear functions, you have to first restrict the domain to find its inverse.

𝑦=𝑥2

𝑥=𝑦 2

±√𝑥=𝑦𝑦=√𝑥   or   𝑦=−√𝑥

Domain:

Inverses of Nonlinear Functions

For some nonlinear functions, you have to first restrict the domain to find its inverse.

𝑦=𝑥2

𝑥=𝑦 2

±√𝑥=𝑦𝑦=√𝑥   or   𝑦=−√𝑥

Domain:

Exercise 6

Find the inverse of the given function.

1. , 2.

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Graphing Investigation

Suppose we drew a triangle on the coordinate plane. Geometrically speaking, what would happen if we switched the x- and the y-coordinates?(𝑥 , 𝑦 )→ (𝑦 , 𝑥 )

(𝑥 , 𝑦 )→ (𝑦 , 𝑥 )

Graphing Investigation

Suppose we drew a triangle on the coordinate plane. Geometrically speaking, what would happen if we switched the x- and the y-coordinates?This is what happens

with inverses.

Graphs of Inverse Functions

Therefore, the graphs of inverse functions are reflections across the line .

Furthermore, to graph , first graph and then switch the - and - coordinates of some key points.

Exercise 7

The graph shows . Graph the inverse of . Is the inverse a function?

Exercise 8

Let f(x) = ½ x – 5.

1. Find f -1

2. State the domain of each function

3. Graph f and f -1 on the same coordinate plane

Exercise 9a

1. For an input of 2, what is the output? Is it unique?

2. For an output of 8, what was the input? Is it unique?

3. What does the answer to Q2 tell you about the inverse of the function?

Exercise 9b

Let f(x) = x3.

1. Find f -1

2. State the domain of each function

3. Graph f and f -1 on the same coordinate plane

Exercise 10a

1. For an input of 4, what is the output? Is it unique?

2. For an output of 16, what was the input? Is it unique?

3. What does the answer to Q2 tell you about the inverse of the function?

Exercise 10b

Let f(x) = x2.

1. Find f -1

2. State the domain of each function

3. Graph f and f -1 on the same coordinate plane

Does it Function?

As the previous Exercise demonstrated, even though you can find the inverse of a function, the inverse itself may not be a function.

Remember, we overcome this shortcoming by restricting the domain of the original function.

Inverses of Nonlinear Functions

For some nonlinear functions, you have to first restrict the domain to find its inverse.

?

𝑦=𝑥2

𝑥=𝑦 2

±√𝑥=𝑦𝑦=√𝑥   or   𝑦=−√𝑥

Domain:

Inverses of Nonlinear Functions

For some nonlinear functions, you have to first restrict the domain to find its inverse.

𝑦=𝑥2

𝑥=𝑦 2

±√𝑥=𝑦𝑦=√𝑥   or   𝑦=−√𝑥

Domain:

Inverses of Nonlinear Functions

For some nonlinear functions, you have to first restrict the domain to find its inverse.

𝑦=𝑥2

𝑥=𝑦 2

±√𝑥=𝑦𝑦=√𝑥   or   𝑦=−√𝑥

Domain:

Does it Function?

Recall that we can use the vertical line test to see if a graph represents a function.

Function Not a Function

The question is: How can we tell if a function’s inverse will be a function?

Horizontal Line Test

The inverse of a function f is also a function iff no horizontal line intersects

the graph of f more than once.

Exercise 11

Graph the function f. Then use the graph to determine whether f -1 is a function.

1. 2. 3.

One-to-One Function

If f passes both the vertical and the horizontal line tests—that is, if f and f -1 are functions—then f is a one-to-one function.

Every input has exactly one output

Every output has exactly one input

Objective 3

You will be able to graph

square and cube root

functions as

transformations on

parent functions

Radical Parents

Square root parent function:

Cube root parent function:

We can perform transformations on these parent functions to help graph whole families of radical functions.

Radical Parents

Use the following to discover the roll of a, h, and k in the following radical functions:

aScaling• 0 < |a| < 1: Shrink vertically• |a| > 1: Stretch vertically• a < 0: Flips

h Horizontal translation

k Vertical translation

𝑦=𝑎√𝑥−h+𝑘𝑦=𝑎 3√𝑥−h+𝑘

Exercise 12

Graph the following radical function. Then state the domain and range.

𝑦=3 √𝑥+2−4

Exercise 13

Graph the following radical function. Then state the domain and range.

𝑦=−123√𝑥−2+3

Exercise 14

Graph the following radical functions. Then state the domain and range.

1. 2.

𝑦=−4 √𝑥−1+2

Inverse FunctionsGraph Square/Cube Root Functions

Objectives:

1. To find the inverse of a function

2. To graph inverse functions

3. To graph square and cube root functions as transformations on parent functions