Functions [parent funct, tranfm, operations, inverse]

Functions

Parent Functions, Transformations, Operations, Inverse and composition.

Unit Review...

Multiple Select

What is a function?

• Equations that can be graphed

• Equations that can be graphed and would pass the vertical line test (intersecting a vertical line at only one point at a time)

• Equations that can be graphed and when viewing ordered pairs, every x input has exactly one y output.

• Equations that can be graphed and passes the horizontal line pass (intersecting a horizontaly line at only one point at a time)

Functions are:

Equations that when graphed would pass the vertical line test.

This means if you pass a vertical line along the graph, the line never intersects the graph more than 1 time.

Also, if you look at the ordered pairs, you notice each value for x has exactly 1-value for y.

Example of Function: {(-2, 4)(-3, 6)(-4, 8)

Example of NOT a function: {(-2, 4), (-3, 6)(-2, 8)}

Multiple Select

Determine which of the following are functions (use your graphing calculator and  the vertical line test). Check all that apply

•  $y=\ 2x^2-3x+4$  

•  $y=\ 3x^{-3^{ }}+3x^2$  

•  $y\ =\ 3x^{3\ }-\ 5x\ +\ 3$  

•  $y\ =\ \frac{3}{x+2}-1$  

If you have questions regarding what a function IS or IS not, how to determine what a function is, write your question down on the attached google document, please and I will answer next class period.

If you have questions about using the vertical line test, please ask.

Multiple Choice

Given:

 $f\left(x\right)=\ 3x^2\ -\ 5$  

Explanin the transformations

• Up 3 units, down 5 units

• left 3 units, down 5 units

• change of size 3 units, down 5 units

• x-axis reflection, down 5 units

Transformation

• Add or Subtract outside of any grouping signs, this is a vertical (up and down) move

   $f\left(x\right)\ =\ \sqrt{x}\ \ +\ 3$         

• Add or subtract inside of any grouping signs, this is a horizontal (left and right)   $h\left(x\right)\ =\ \left|x\ -\ 3\right|$          

• Negative out front, x-axis reflection   $h\left(x\right)\ =\ \ -\ x^3$  

• Negative inside grouping symbols, y-axis reflection    $h\left(x\right)\ =\ \left(-x\right)^3$  

• Number multiplied to x, change in size.   $k\left(x\right)\ =\ 4x^2$  or   $k\left(x\right)\ =\ \left(3x\right)^2$  

Examples

• y = 4x² - 2, change in size by factor of 4, shift down 2 units

• y = (x + 3)³ - 4 , of 3 units left and 4 units down

• y = -x³ -2, x-axis reflection shift 2 units down

• y = (-4x)³, is change in size by factor of 4, y-axis reflection

• Questions? Write them on the google document.

Multiple Choice

The blue function is the original function f(x) = x³. Which of the following is the correct equation for the red function, g(x)?

• g(x)=x³+1

• g(x)=x³-1

• g(x)=(x-1)³

• g(x)=(x+1)³

Multiple Choice

f(x) = ⅔(x - 7)²

• Shrink of ⅔

  Right 7

• Shrink of ⅔

  Left 7

• Stretch of ⅔

  Right 7

• Stretch of ⅔

  Left 7

Multiple Choice

Match the equation to its description.

• Right 2 and up 2

• Left 2 and up 2

• Right 2 and down 2

• Left 2 and down 2

Multiple Choice

Which equation describes the transformation of shifting f(x)=x³ three units left?

• x³ + 3
• x³ - 3
• (x + 3)³
• (x - 3)³

Multiple Choice

Which equation describes the transformation of shifting f(x)=x³ seven units up?

• x³+7

• x³-7

• (x-7)³

• (x+7)³

Multiple Choice

.

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

• g(x)=(x+3)²

• g(x)=(x-3)²

• g(x)=x²+3

• g(x)=x²-2

Multiple Choice

f(x) = 5x² + 2

• Stretch of 5

  Up 2

• Shrink of 5

  Up 2

• Stretch of 5

  Down 2

• Shrink of 5

  Down 2

Multiple Choice

f(x) = -(4x - 2)³ + 3

• x-axis reflection, change in size by factor of 3, right 2, up 4

• y-axis reflection, change in size by factor of 4, right 2, up 3

• x-axis reflection, change in size by factor of 4, right 2, up 3

• x-axis reflection, change in size by factor of 4, left 2, up 3

Operations with Functions

Operations with functions is adding, subtracting, multiplying or dividing functions.

If you have questions, please have them ready.

Multiple Choice

All of the y values or outputs are called what?

• Domain
• Range
• Relation
• Function

Multiple Choice

f(x) = 3 and g(x) = -2x + 5

Find f(x) * g(x)

• 6x + 15

• 6x - 15

• -6x + 15

• 6x - 8

There are 4 different types of function operations divided into 2 groups.

• Type 1: Simple evaluate. This question will look like: f(x)=4x - 2, find f(3). This means to replace x with 3--> f(x) 4x - 2 --> f(3)=4(3) - 2=10

• Type 2: Evaluate 2 functions and add/subtract/multiply or divided. This questions will look like: f(x) = 2x - 1 and g(x) = 10x+2, find (f+g)(3) or it may look like f(3) + g(3), both mean to replace x with 2 and add: 2(3) - 1 + 10(3) + 2 = 5 + 32 3

• Type 3: Evaluate a function with an expression. This question will look like; j(x) = 3x - 1, find j (x²+6). This is similar to the above, but instead of replacing x with number, you replace x with the expression: j(x) = 3x - 1, j (x²+6) 3( x²+6 ) - 1 = 3x² + 18 - 1 = 3x² + 17

^

• Type 4: Evaluating an expression with more than 1 function:

• f(x) = 7x - 2, g(x) = x+2, find (f - g)(2x² + 1)

• [7x - 2] - [x+2]

• [7(x² + 1) - 2] - [(x² + 1)+2]

• (7x² + 7) - (x²+1+2)

• 6x² + 4

Multiple Choice

f(n)=n²+4n

g(n)=-n-5

Find f(n)+g(n)

• n²+3n-5

• -n³+2n-7

• n³+3n-3n

• n²-3n-5

Multiple Choice

f(n)=x²+1

g(n)=2x-5

Find f(n)-g(n)

• x²-2x+6

• -x²+2x+4

• -x²-2x-6

• x²-2x-4

Multiple Choice

Which equation represents the pattern shown in the table?

• y = ¹/₂x + 2

• y = ¹/₂x - 2

• y = 2x - 2

• y = 2x + 2

Multiple Choice

For what value of x does f(x) = -2?

• 0

• -5

• 3

• -3

Multiple Choice

What is f(2)?

• -4
• 8
• 0
• 5

Multiple Choice

 $Given\ h\left(x\right)=3x^2-2x+8\ what\ is\ h\left(2\right)?$  

• 13

• 8

• 16

• answer not here

Multiple Choice

$f\left(x\right)=5x-12$ What is $f\left(-4\right)$

• -32

• -3

• -12

• 3

Multiple Choice

If a question asks you to find f(2), what does that mean?

• Let x = 2

• Let y = 2

Multiple Choice

f(x) is the same thing as ______.

• y

• x

• input

• an equation

Multiple Choice

f(x) is the same thing as ______.

• y

• x

• input

• an equation

Multiple Choice

All of the x values or inputs are called what?

• Domain
• Range
• Relation
• Function

Inverse of a function

This is the REFLECTION of the function over the line y = x

The function and it's inverse ordered pairs will be mirror images of each other.

For example f(x) might have the order pairs {(-4, 2)( - 5, 3)(-6, 2)(-7,1)}

f^-1(x) would have order pairs of

{2, -4). (3, -5) (2, -6). (1, -7)

Notice how ALL of the ordered pairs have been reversed or are mirror images

Multiple Choice

What is the inverse of the given coordinates: (1, 2) (3, 8) (-3, 6)

• (1, 2) (3, 8) (-3, 6)

• (1, 2) (8, 3) (-3, 6)

• (-1, -2) (-3, -8) (3, -6)

• (2, 1) (8, 3) (6, -3)

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

What does it mean to find the inverse of a function?

• the x's and y's are switched & its a reflection over the x-axis

• the x's and y's are switched & its a reflection over the y-axis

• the x's and y's are switched & its a reflection over a line

• the x's and y's are switched & its a reflection over the line y = x

Multiple Choice

If the equation of f(x) goes through (1, 4) and (4, 6), what points does the inverse: f^-1(x) go through?

• (1, 4) and (4, 6)

• (-4, -1) and (-6, -4)

• (-1, -4) and (-4, -6)

• (4, 1) and (6, 4)

Multiple Choice

If the equation of f(x) goes through (1, 4) and (4, 6), what points does f^-1(x) go through?

• (1, 4) and (4, 6)
• (-4, -1) and (-6, -4)
• (-1, -4) and (-4, -6)
• (4, 1) and (6, 4)

Multiple Choice

Find the inverse of the function
f(x) = 3x − 5

• f^-1(x)=3x +5
• f^-1(x)=(x+5)/3
• f^-1(x)=3y-5
• f^-1(x)=3y+5

Multiple Choice

Find the inverse of f(x) = -4x - 12

• f^-1(x) = 4x - 3

• f^-1(x) = -1/4x - 3

• f^-1(x) = 1/4x + 3

• f^-1(x) = -4x - 3

Multiple Choice

Find the inverse of f(x) = -4x - 12

• f^-1(x) = 4x - 3

• f^-1(x) = -1/4x - 3

• f^-1(x) = 1/4x + 3

• f^-1(x) = -4x - 3

How to write the inverse for a function

Step 1: replace f(x) with y

Step 2: swap the place for x and y (put y where x is and x where y is)

Step 3: Resolve for y

Step 4: Verify they ARE inverses

Step 5: Replace y with f^-1(x) {use what ever function notation from the problem, f(x) -->f^-1(x); h(x) -->h^-1(x): g(x) -->g^-1(x)

THE END

You can attempt this 5 times

This can be used to study for the upcoming test

REMEMBER, write your questions DOWN on the document!