functions review. jeopardy! definitions rational and equivalent functions functions transformations...

Functions Review

JEOPARDY!Definiti

ons

Rational and Equivalent Functions

Functions Transformations

Inverse Functions

10 10 10 10 10

20 20 20 20 20

30 30 30 30 30

40 40 40 40 40

50 50 50 50 50

Definitions - 10

Define the domain and range of a function.

Definitions – 10Answer

The domain of a function is the set of all x values that produce a y value.

The range of a functions is set of all y values that result from any function.

Definitions - 20

List all forms (3) of a quadratic function.


• Standard• Factored• Vertex

Definitions - 30

In what order should transformations be completed?


• Horizontal reflection and stretch• Horizontal shift• Vertical reflection and stretch• Vertical shift

Definitions - 40

What line is an inverse functions reflected in?


The line y = x

Definitions - 50

List all possible transformations (6) and their corresponding letter.


k – reflection if k is negativek – horizontal stretch by a factor of 1/kp – horizontal shifta – reflection if a is negativea – vertical stretch by a factor of aq – vertical shift

Rational and Equivalent Functions - 10

Are the two expressions equivalent?

(x-3)2(x+4)(x+1)

x4- x3 -17x2 +21x+36

Rational and Equivalent Functions – 10Answer

Yes

Solution: Test at least 3 points in both functions to make sure they are they return the same value.

For example, test x=0

Function 1: (x-3)2(x+4)(x+1)=(0-3) 2(0+4)(0+1)=(-3) 2(4)(1)=36

Function 2: x4- x3 -17x2 +21x+36=04-03-17(0) 2+21(0)+36=36


Simplify the function and state all restrictions.

-x2 -7x+8 . x +5 x+8 9x-9


Solution: -x2 -7x+8 . x +5 x+8 9x-9= -(x-1)(x+8) . x+5 x+8 9(x-1)= -(x+5) 9

-(x+5) 9

x ≠ -8, 1


Write the function in standard form.

y = 2(x-4)2 -7


Solution:y = 2(x-4)2 -7 = 2(x2 -8x+16) -7 = 2x2 -16x+32-7 = 2x2 -16x+25

y = 2x2 -16x + 25



x+3 ÷ (x-1)(x+3)x+2 (x-1)2


Solution:x+3 ÷ (x-1)(x+3)x+2 (x-1)2

=x+3 . (x-1)2__ x+2 (x-1)(x+3)= x-1 x+2

x - 1 x+ 2

x ≠ -2, 1, -3



5 - x _ x2 -5x 5x-25


Solution: 5 - x _ x2 -5x 5x-25= 5 - x x(x-5) 5(x-5)= 5 . 5- x . x x(x-5) 5 5(x-5) x= 25 - x2 5x(x-5) 5x(x-5)= 25-x2

5x(x-5)= - (x2-25) 5x(x-5)= -(x+5)(x-5) 5x(x-5)= -(x+5) 5x

Find a common denominator: 5x(x-5)

-(x+5) 5x

x ≠ 0, 5

Functions - 10

State the restrictions of y = √x-4 .___

Functions – 10Answer

x ≥ 4

Functions - 20

What is the function for the following graph?


y = √x_

Functions - 30

State the domain and range of y = (x+2)2


Domain: {x ε R}Range: {y ε R | y ≥ 0}

Functions - 40

List all base points of f(x) = 1/x


(-2, -1/2)(-1, -1)(-1/2, -2)(1/2, 2)(1, 1)(2, 1/2)

Functions - 50

Write the general form of a transformed function.


y = af(k(x-p))+q

Transformations - 10

If you start at a point (3, 5) and move 4 units left and 3units up, what is the new coordinate?

Transformations – 10Answer

(-1, 8)


List the transformations on the function.

f(x) = -3(x-5)2+1


• Shift 5 units right• Reflection in the x-axis (vertical reflection)• Vertical stretch by a factor of 3• Shift 1 unit up


Choose any three base points and write them after thetransformation.

f(x) = 5√-2(x+1) ______


(0, 0) (-1, 0) (1, 1) (-1.5, 5) (4, 2) (-5, 10) (9, 3) (-5.5, 15) (16, 4) (-9, 20)

(x, y) (- x, y) (- x-1, y) (-2s+1, 5y)

(0, 0) (0, 0) (-1, 0) (-1, 0)(1, 1) (-0.5, 1) (-1.5, 1) (-1.5, 5)(4, 2) (-4, 2) (-5, 2) (-5, 10)(9, 3) (-4.5, 3) (-5.5, 3) (-5.5, 15)(16, 4) (-8, 4) (-9, 4) (-9, 20)


List all the transformations on the function.


• Shift 4 units right• Vertical reflection• Shift 6 units up


Write the base function and the transformed function.


Base function: y = 1/x

Transformed function: y = - 1/(x+3) -2

Inverse Functions - 10

What is the inverse of {(-7, 12), (2, 0), (-10, 4)}.

Inverse Functions – 10Answer

{(0, 2), (4, -10), (12, -7)}


What is the inverse of y = (1/3 )x +4


f -1(x)= 3x-12

Solution:y = (1/3 )x +4x = (1/3 )y+4x-4 = (1/3 )y3(x-4) =y

f -1 (x) = 3(x-4) = 3x-12


What is the inverse of f(x) = 2√-x +3 __


f -1(x)= -(1/4)(x – 3)2

Solution:f(x) = 2√-x +3y = 2√-x +3x = 2√-y+3x-3 = 2√-yx-3 = √-y 2x-3 2 = -y 2 x-3 2 = y 2

f -1(x) = x-3 2

2

( )

-( )

-( )

______

____


What is the inverse of f(x) = 2x2+16x+29


f -1(x)= 4±√(1/2)(x+3)

Solution:f(x) = 2x2+16x+29y = 2x2+16x+29y = 2(x2+8x) +29y = 2(x2 +8x +16-16) +29y = 2(x+4) 2 -32+29y = 2(x+4) 2 -3x = 2(y+4) 2 -3X+3 = 2(y+4) 2 X+3 = (y+4) 2 2X+3 = y+4 2X+3 -4 = y 2

________

±√__

__±√ f -1 (x) =-4 x+3

2±√ __


Sketch the original and the inverse of the followingfunction (without finding the inverse). Is the inverse afunction?

HINT - draw the y=x line and then draw the reflection

f(x) = -(3(x-5))2 -1


Solution:Pick at least three points on the original function then use them to get three pointsfor the inversefunction.

*Make sure that sketches aresomewhataccurate (i.e. the vertex shouldbe at the correctpoint *