Meera practice

Inverse Functions

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Inverse Functions

Coordinates

• f(x) has the coordinates (2,9) , (-2,5), and (4, -6)

• The inverse, f^-1(x) has the coordinates (9,2) , (5, -2) and (-6,4)

• notice how the x's and y's switched

• f(x) --> (x,y)

• f^-1(x) --> (y,x)

Graphing

• When we graph the function an the inverse, we see that the inverse if a reflection over the line y=x

• The dotted line on the graph is the line y=x

Multiple Choice

What are the inverse points for the coordinates:

(3, 0) , (2, -4), (5, 5) and (-6,8)

• (3,0) , (2, -4) , (5,5) , (6,8)

• (0,3) , (-4,2) , (5,5) , (8,-6)

• (3,2), (0, -4), (5, -6), (5,8)

Multiple Choice

Are these inverse functions?

• yes

• no

Multiple Choice

Are these functions inverses?

• No

• Yes

• No way to tell

Multiple Choice

Are these functions inverses?

• Yes

• No

Multiple Choice

The inverse has been reflected over which line?

• y = x
• y =1
• y = 0
• y = x + 1

Multiple Choice

Which graph represents the functions f(x) and f^-1(x)?

Finding an Inverse

• Step 1: change f(x) to y

• Step 2: switch x and y

• Step 3: solve for y

• Step 4: change y to f^-1(x)

Multiple Choice

Find the inverse of  $f(x)=10+7x$  

• $f^{-1}(x)=\frac{x-7}{10}$

• $f^{-1}(x)=\frac{x-10}{7}$

• $f^{-1}(x)=\frac{1}{10+7x}$

Multiple Choice

Find the inverse of  $f(x)=4x$  

• $f^{-1}(x)=\frac{x}{4}$

• $f^{-1}(x)=4$

• $f^{-1}\left(x\right)=\frac{1}{4}$

Multiple Choice

Find the inverse of  $f(x)=\frac{x-5}{3}$  

• $f^{-1}(x)=\frac{3}{5}+x$

• $f^{-1}(x)=3x+5$

• $f^{-1}(x)=5x+3$

Multiple Choice

Find the inverse of  $f(x)=2x+1$  

• $f^{-1}(x)=2x$

• $f^{-1}(x)=\frac{x-1}{2}$

• $f^{-1}(x)=2$

Multiple Choice

Find the inverse of  $f(x)=3x+2$  

• $f^{-1}(x)=3x-2$

• $f^{-1}(x)=2+3x$

• $f^{-1}(x)=\frac{x-2}{3}$

Multiple Choice

The inverse of $f(x)=8x+30$  is  $f^{-1}(x)=\frac{8}{30}$  .

• True

• False

Multiple Choice

The inverse of $f(x)=x+30$   is   $f^{-1}(x)=x-30$  .

• True

• False

• I have no idea how to do this.

Multiple Choice

The inverse of the function f(x) is written as ...

• f ^-1(x)

• f ²(x)

• f '(x)

• f ⁺(x)

​4 steps to find inverse function

•Step1: Let f(x) = y

•Step 2: Interchange x and y

•Step 3: Solve for y

•Step 4: Change y to f^-1(x)

Subject | Subject

Some text here about the topic of discussion

​Find the inverse of the following function: f(x) = x -3

​4 steps to find inverse function

•Step1: Let f(x) = y

•Step 2: Interchange x and y

•Step 3: Solve for y

•Step 4: Change y to f^-1(x)

Subject | Subject

Some text here about the topic of discussion

​Find the inverse of the following function: f(x) = x -3

​4 steps to find inverse function

•Step1: Let f(x) = y

•Step 2: Interchange x and y

•Step 3: Solve for y

•Step 4: Change y to f^-1(x)

Subject | Subject

Some text here about the topic of discussion

​Find the inverse of the following function: f(x) = 2x² + 3

​4 steps to find inverse function

•Step1: Let f(x) = y

•Step 2: Interchange x and y

•Step 3: Solve for y

•Step 4: Change y to f^-1(x)

Subject | Subject

Some text here about the topic of discussion

​Find the inverse of the following function: f(x) = 2x² + 3

Horizontal Line Test

This test is used to find one to one functions

VERTICAL LINE TEST

The vertical line test is used to determine if a graph is a function or not.

If the verticle line hits the graph ONE time IT IS a function.

If the verticle line hits the graph TWO or MORE times it is NOT a function.

Multiple Choice

Identify the type of function represented by the graph

• Identity/Linear

• Constant

• Quadratic

• Absolute Value

Multiple Choice

Identify the type of function represented by the graph

• Identity/Linear

• Constant

• Quadratic

• Absolute Value

Multiple Choice

Identify the type of function represented by the graph

• Identity/Linear

• Constant

• Quadratic

• Absolute Value

Multiple Choice

Identify the type of function represented by the graph

• Identity/Linear

• Constant

• Quadratic

• Absolute Value

Multiple Choice

Describe the translation in y = |x – 4|. Then graph the function.

• translation of the graph

  y = |x| up 4 units

• translation of the graph

  y = |x| down 4 units

• translation of the graph

  y = |x| right 4 units

• translation of the graph

  y = |x| left 4 units

Multiple Choice

Describe the dilation in y = |2x|. Then graph the function.

• dilation of the graph of y = |x| compressed vertically

• dilation of the graph of y = |x| stretched vertically

• dilation of the graph of y = |x| translated 2 units up

• dilation of the graph of y = |x| translated 2 units right

Multiple Choice

Describe the transformation from the parent absolute value function:

$$y=\left|x\right|-5$$  

• Shift up 5 units

• Shift down 5 units

• Shift right 5 units

• Shift left 5 units

​Horizontal Translation

This allows our function to move LEFT or RIGHT

​

​** These are inside the parentheses **

​

(x – h) shifts right

(x + h) shifts left

Multiple Choice

Describe the transformation from the parent graph

• Reflect over x, right 3, down 4

• Reflect over y, right 3, down 4

• Reflect over x, right 3, up 4

• Reflect over y, left 3, down 4

Multiple Choice

If the blue is f(x)=x^2, then the red must be

• g(x) = x² - 5

• g(x) = x² + 5

• g(x) = (x - 5)²

• g(x) = (x + 5)²

Multiple Choice

f(x) is transformed so that it maps onto g(x). Which equation would correctly represent g(x)

• g(x) = x² - 3

• g(x) = x² + 3

• g(x) = (x - 3)²

• g(x) = (x + 3)²

Multiple Choice

The vertex of the quadratic function f(x) in the picture is (-4, 5).

If g(x) = f(x) translated LEFT 2 units , what is the vertex of g(x)?

• (-2, 5)
• (-6, 5)
• (-4, 3)
• (-4, 7)

Multiple Choice

y = x² + 2

• LEFT 2

• RIGHT 2

• UP 2

• DOWN 2

Multiple Choice

y = |x|- 4

• UP 4

• DOWN 4

• LEFT 4

• RIGHT 4

Multiple Choice

y = |x - 6|

• UP 6

• DOWN 6

• LEFT 6

• RIGHT 6

Multiple Choice

y = (x + 4)³

• UP 4

• DOWN 4

• LEFT 4

• RIGHT 4

Horizontal Shifts

This allows our function to move LEFT and RIGHT or horizontally. This occurs INSIDE parenthesis.

Transformations of Parent Functions Review

​

Let's Review Our Rules!

Multiple Choice

y = x² + 2

• LEFT 2

• RIGHT 2

• UP 2

• DOWN 2

Multiple Choice

y = |x|- 4

• UP 4

• DOWN 4

• LEFT 4

• RIGHT 4

Vertical Shifts

This allows our function to move UP and DOWN or vertically. These are outside of the parenthesis!!

Multiple Choice

y = |x - 6|

• UP 6

• DOWN 6

• LEFT 6

• RIGHT 6

Multiple Choice

y = (x + 4)³

• UP 4

• DOWN 4

• LEFT 4

• RIGHT 4

Horizontal Shifts

This allows our function to move LEFT and RIGHT or horizontally. This occurs INSIDE parenthesis.

Multiple Choice

y = - (x)²

• Stretch by -1

• Reflection over x - axis

Reflections

When there is a negative sign in front of your equation this means to reflect over the X - AXIS!

Multiple Choice

y = 3(x)²

• Stretch by factor of 3

• Compress by factor of 3

• Stretch by factor $\frac{1}{3}$

• Compress by factor of $\frac{1}{3}$

Multiple Choice

y = $\frac{1}{1000}$  ( $x^2$ )

• Stretch by factor of   $\frac{1}{1000}$ 

• Compress by factor of  $\frac{1}{1000}$  

• Stretch by factor of 1000

• Compress by factor of 1000

Stretch & Compress

If a > 1 : There is a STRETCH

If 0 < a < 1 : There is a COMPRESSION

Multi-Step Transformations

Let's put it all together!!!!! So FUN!

​

Multiple Choice

y = (x - 4)² - 7

• Down 4, Down 7

• Down 4, Up 7

• Right 4, Down 7

• Left 4, Down 7

Multiple Choice

y = - (x)³ + 10

• Reflect over x - axis, down 10

• Reflect over x - axis, up 10

• Reflect over x - axis, Left 10

• Reflect over x - axis, Right 10

Multiple Choice

y = - 6|x - 8|

• Stretch by - 6

• Compress by - 6

• Reflect over x - axis, stretch 6

• Reflect over x - axis, compress by 6

Multiple Choice

y = - (x + 5)² + 5

• Reflect over x - axis, Right 5, Up 5

• Reflect over x - axis, Left 5, Down 5

• Reflect over x - axis, Right 5, Down 5

• Reflect over x - axis, Left 5, Up 5

Multiple Select

Check all that apply:

y = - 7 (x + 9)³

• Reflect over x - axis

• Compress by -7

• Up 9

• Left 9

• Stretch by 7

Multiple Select

 Check all that apply:
y =  $\frac{3}{8}$ |x - 4| + 1

• Compress by  $\frac{3}{8}$  

• Left 4

• Down 1

• Right 4

• Up 1

Multiple Select

Check all that apply:

y = -4 (x - 7)² + 6

• Compress by -4

• Stretch by 4

• Right 7

• Up 6

• Reflect over x - axis

Transformations of Parent Functions Review

​

Let's Review Our Rules!

Multiple Choice

y = x² + 2

• LEFT 2

• RIGHT 2

• UP 2

• DOWN 2

Multiple Choice

y = |x|- 4

• UP 4

• DOWN 4

• LEFT 4

• RIGHT 4

Vertical Shifts

This allows our function to move UP and DOWN or vertically. These are outside of the parenthesis!!

Multiple Choice

y = |x - 6|

• UP 6

• DOWN 6

• LEFT 6

• RIGHT 6

Multiple Choice

y = (x + 4)³

• UP 4

• DOWN 4

• LEFT 4

• RIGHT 4

Horizontal Shifts

This allows our function to move LEFT and RIGHT or horizontally. This occurs INSIDE parenthesis.

Multiple Choice

y = - (x)²

• Stretch by -1

• Reflection over x - axis

Reflections

When there is a negative sign in front of your equation this means to reflect over the X - AXIS!

Multiple Choice

y = 3(x)²

• Stretch by factor of 3

• Compress by factor of 3

• Stretch by factor $\frac{1}{3}$

• Compress by factor of $\frac{1}{3}$

Multiple Choice

y = $\frac{1}{1000}$  ( $x^2$ )

• Stretch by factor of   $\frac{1}{1000}$ 

• Compress by factor of  $\frac{1}{1000}$  

• Stretch by factor of 1000

• Compress by factor of 1000

Stretch & Compress

If a > 1 : There is a STRETCH

If 0 < a < 1 : There is a COMPRESSION

Multi-Step Transformations

Let's put it all together!!!!! So FUN!

​

Multiple Choice

y = (x - 4)² - 7

• Down 4, Down 7

• Down 4, Up 7

• Right 4, Down 7

• Left 4, Down 7

Multiple Choice

y = - (x)³ + 10

• Reflect over x - axis, down 10

• Reflect over x - axis, up 10

• Reflect over x - axis, Left 10

• Reflect over x - axis, Right 10

Multiple Choice

y = - 6|x - 8|

• Stretch by - 6

• Compress by - 6

• Reflect over x - axis, stretch 6

• Reflect over x - axis, compress by 6

Multiple Choice

y = - (x + 5)² + 5

• Reflect over x - axis, Right 5, Up 5

• Reflect over x - axis, Left 5, Down 5

• Reflect over x - axis, Right 5, Down 5

• Reflect over x - axis, Left 5, Up 5

Multiple Select

Check all that apply:

y = - 7 (x + 9)³

• Reflect over x - axis

• Compress by -7

• Up 9

• Left 9

• Stretch by 7

Multiple Select

 Check all that apply:
y =  $\frac{3}{8}$ |x - 4| + 1

• Compress by  $\frac{3}{8}$  

• Left 4

• Down 1

• Right 4

• Up 1

Multiple Select

Check all that apply:

y = -4 (x - 7)² + 6

• Compress by -4

• Stretch by 4

• Right 7

• Up 6

• Reflect over x - axis

Vertical Shifts

This allows our function to move UP and DOWN or vertically. These are outside of the parenthesis!!

Reflections

When there is a negative sign in front of your equation this means to reflect over the X - AXIS!

Multiple Choice

y = 3(x)²

• Stretch by factor of 3

• Compress by factor of 3

• Stretch by factor $$\frac{1}{3}$$

• Compress by factor of $$\frac{1}{3}$$

Stretch & Compress

If a > 1 : There is a STRETCH

If 0 < a < 1 : There is a COMPRESSION

Multi-Step Transformations

Let's put it all together!!!!! So FUN!

Multiple Choice

y = (x - 4)² - 7

• Down 4, Down 7

• Down 4, Up 7

• Right 4, Down 7

• Left 4, Down 7

Multiple Choice

y = - (x)³ + 10

• Reflect over x - axis, down 10

• Reflect over x - axis, up 10

• Reflect over x - axis, Left 10

• Reflect over x - axis, Right 10