Functions Review

Rational and Equivalent Functions JEOPARDY! Definitions Rational and Equivalent Functions Functions Transformations Inverse Functions 10 20 30 40 50

Definitions - 10 Define the domain and range of a function.

Definitions – 10 Answer 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.

Definitions – 20 Answer Standard Factored Vertex

Definitions - 30 In what order should transformations be completed?

Definitions – 30 Answer Horizontal reflection and stretch Horizontal shift Vertical reflection and stretch Vertical shift

Definitions - 40 What line is an inverse functions reflected in?

Definitions – 40 Answer The line y = x

Definitions - 50 List all possible transformations (6) and their corresponding letter.

Definitions – 50 Answer k – reflection if k is negative k – horizontal stretch by a factor of 1/k p – horizontal shift a – reflection if a is negative a – vertical stretch by a factor of a q – 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 – 10 Answer 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

Rational and Equivalent Functions - 20 Simplify the function and state all restrictions. -x2 -7x+8 . x +5 x+8 9x-9

Rational and Equivalent Functions – 20 Answer 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) x ≠ -8, 1

Rational and Equivalent Functions - 30 Write the function in standard form. y = 2(x-4)2 -7

Rational and Equivalent Functions – 30 Answer Solution: y = 2(x-4)2 -7 = 2(x2 -8x+16) -7 = 2x2 -16x+32-7 = 2x2 -16x+25 y = 2x2 -16x + 25

Rational and Equivalent Functions - 40 Simplify the function and state all restrictions. x+3 ÷ (x-1)(x+3) x+2 (x-1)2

Rational and Equivalent Functions – 40 Answer 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

Rational and Equivalent Functions - 50 Simplify the function and state all restrictions. 5 - x _ x2 -5x 5x-25

Rational and Equivalent Functions – 50 Answer 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) = -(x+5)(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 – 10 Answer x ≥ 4

Functions - 20 What is the function for the following graph?

Functions – 20 Answer _ y = √x

Functions - 30 State the domain and range of y = (x+2)2

Functions – 30 Answer Domain: {x ε R} Range: {y ε R | y ≥ 0}

Functions - 40 List all base points of f(x) = 1/x

Functions – 40 Answer (-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.

Functions – 50 Answer y = af(k(x-p))+q

Transformations - 10 If you start at a point (3, 5) and move 4 units left and 3 units up, what is the new coordinate?

Transformations – 10 Answer (-1, 8)

Transformations - 20 List the transformations on the function. f(x) = -3(x-5)2+1

Transformations – 20 Answer Shift 5 units right Reflection in the x-axis (vertical reflection) Vertical stretch by a factor of 3 Shift 1 unit up

Transformations - 30 Choose any three base points and write them after the transformation. ______ f(x) = 5√-2(x+1)

Transformations – 30 Answer (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) (-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)

Transformations - 40 List all the transformations on the function.

Transformations – 40 Answer Shift 4 units right Vertical reflection Shift 6 units up

Transformations - 50 Write the base function and the transformed function.

Transformations – 50 Answer 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 – 10 Answer {(0, 2), (4, -10), (12, -7)}

Inverse Functions - 20 What is the inverse of y = (1/3 )x +4

Inverse Functions – 20 Answer f -1(x)= 3x-12 Solution: y = (1/3 )x +4 x = (1/3 )y+4 x-4 = (1/3 )y 3(x-4) =y f -1 (x) = 3(x-4) = 3x-12

Inverse Functions - 30 __ What is the inverse of f(x) = 2√-x +3

Inverse Functions – 30 Answer f -1(x)= -(1/4)(x – 3)2 Solution: f(x) = 2√-x +3 y = 2√-x +3 x = 2√-y+3 x-3 = 2√-y x-3 = √-y 2 x-3 2 = -y x-3 2 = y f -1(x) = x-3 2 __ __ __ __ __ ( ) -( ) -( )

Inverse Functions - 40 What is the inverse of f(x) = 2x2+16x+29

Inverse Functions – 40 Answer ________ f -1(x)= 4±√(1/2)(x+3) Solution: f(x) = 2x2+16x+29 y = 2x2+16x+29 y = 2(x2+8x) +29 y = 2(x2 +8x +16-16) +29 y = 2(x+4) 2 -32+29 y = 2(x+4) 2 -3 x = 2(y+4) 2 -3 X+3 = 2(y+4) 2 X+3 = (y+4) 2 2 X+3 = y+4 X+3 -4 = y __ ±√ __ __ ±√ ±√ f -1 (x) =-4 x+3 2

Inverse Functions - 50 Sketch the original and the inverse of the following function (without finding the inverse). Is the inverse a function? f(x) = -(3(x-5))2 -1 HINT - draw the y=x line and then draw the reflection

Inverse Functions – 50 Answer Solution: Pick at least three points on the original function then use them to get three points for the inverse function. *Make sure that sketches are somewhat accurate (i.e. the vertex should be at the correct point *