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May 2020

May 2020

Modeling with Functions

Function Family

Function Name Algebraic Shape of Graph

Linear 𝑦 = 𝑚𝑥 + 𝑏

Exponential 𝑦 = 𝑎𝑏𝑥

Quadratic 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

Polynomial 𝑦 = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2 + ⋯ 𝑎0

May 2020

Rational 𝑦 =

1

𝑥

Absolute Value 𝑦 = |𝑥|

Logarithmic 𝑦 = 𝑙𝑜𝑔𝑏(𝑥)

Trigonometric 𝑦 = 𝑎 sin(𝑏𝑥)

or

𝑦 = 𝑎 cos (𝑏𝑥)

Radical 𝑦 = √𝑥𝑛

May 2020

Function Transformation Rules

Examples: The parent function is 𝑦 = 𝑥

a. 𝑦 = 𝑥 + 2

the function shifted up by 2

b. 𝑦 = 𝑥 − 2

the function shifted down by 2

c. 𝑦 = 2𝑥

the function compressed horizontally by 2

d. 𝒚 = −𝒙

the function reflected over the y – axis

May 2020

May 2020

Try It:

Graph the following quadratic equations on the grid. The equation 𝑦 = 𝑥2 has been graphed for you. For each new

equation explain what the number 3 does to the graph 𝑦 = 𝑥2. Pay attention to the y-intercept, the x-intercept(s), and

the rate of change. Identify what changes in the graph and what stays the same.

a. 𝑦1 = 𝑥2 + 3

b. 𝑦2 = 𝑥2 − 3

c. 𝑦3 = (𝑥 − 3)2

d. 𝑦4 = (𝑥 + 3)2 + 3

e. 𝑦1 = 3𝑥2

Practice Problems: Sketch the graph of the parent function and the graph of the transformed function on the same set of axes.

1.

2.

3. 4.

May 2020

May 2020

Composing and Decomposing

Composing Functions: Applying one function to the results of another. A composite function is created

when one function is substituted into another function.

It can be written as (𝑓 ○ 𝑔)(𝑥) which means 𝑓(𝑔(𝑥)). This can be read as “ 𝑓 of 𝑔 of 𝑥”

Examples:

1. Given: 𝑓(𝑥) = 2𝑥2 + 1 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 − 4. 𝐹𝑖𝑛𝑑 𝑓(𝑔(𝑥)).

𝑓(𝑔(𝑥)) = 𝑓(𝑥 − 4) [𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑥 − 4 𝑖𝑛𝑡𝑜 𝑔(𝑥)]

= 2(𝑥 − 4)2 + 1 [ 𝐹𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 𝑖𝑛 𝑓(𝑥), 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑥 − 4]

= 2(𝑥2 − 8𝑥 + 16) + 1 [ 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑏𝑦 𝑒𝑥𝑝𝑎𝑛𝑑𝑖𝑛𝑔 (𝑥 − 4)2]

= 2𝑥2 − 16𝑥 + 32 + 1 [ 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒 2]

𝑓(𝑔(𝑥)) = 2𝑥2 − 16𝑥 + 33 [𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦]

2. Given: 𝑓(𝑥) = 4𝑥 + 9 𝑎𝑛𝑑 𝑔(𝑥) =𝑥−9

4. 𝐹𝑖𝑛𝑑 𝑔(𝑓(𝑥)).

𝑔(𝑓(𝑥)) = 𝑔(4𝑥 + 9) [𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 4𝑥 + 9 𝑖𝑛𝑡𝑜 𝑓(𝑥)]

=(4𝑥+9)−9

4 [ 𝐹𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 𝑖𝑛 𝑔(𝑥), 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 4𝑥 + 9]

= 4𝑥

4 [ 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 ]

𝑔(𝑓(𝑥)) = 𝑥 [ 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 ]

3. Given: 𝑓(𝑥) = 4𝑥 + 9 𝑎𝑛𝑑 𝑔(𝑥) =𝑥−9

4. 𝐹𝑖𝑛𝑑 𝑓(𝑓(𝑥)).

𝑓(𝑓(𝑥)) = 𝑓(4𝑥 + 9) [𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 4𝑥 + 9 𝑖𝑛𝑡𝑜 𝑓(𝑥)]

= 4(4𝑥 + 9) + 9 [ 𝐹𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 𝑖𝑛 𝑓(𝑥), 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 4𝑥 + 9]

= 16𝑥 + 36 + 9 [ 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 ]

𝑓(𝑓(𝑥)) = 16𝑥 + 45 [ 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 ]

4. Given: 𝑓(𝑥) = 4𝑥 + 9 𝑎𝑛𝑑 𝑔(𝑥) =𝑥−9

4. 𝐹𝑖𝑛𝑑 𝑓(𝑓(3)).

𝑓(𝑓(𝑥)) = 16𝑥 + 45 [𝑤𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑖𝑠 𝑓𝑟𝑜𝑚 𝑒𝑥𝑎𝑚𝑙𝑒 3]

𝑓(𝑓(3)) = 16(3) + 45 [𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 3 𝑖𝑛𝑡𝑜 𝑥]

= 48 + 45 [ 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 ]

𝑓(𝑓(3)) = 93 [ 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 ]

May 2020

May 2020

Practice Problems:

1. Let 𝑓(𝑥) = 2𝑥2 − 4 𝑎𝑛𝑑 𝑔(𝑥) = 5𝑥. 𝐹𝑖𝑛𝑑 𝑒𝑎𝑐ℎ 𝑎𝑛𝑑 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦.

a) (𝑓 ○ 𝑔)(1) b) (𝑔 ○ 𝑓)(1) c) (𝑓 ○ 𝑓)(−2) d) (𝑔 ○ 𝑔)(−1)

2. Let 𝑓(𝑥) =8

𝑥−3𝑎𝑛𝑑 𝑔(𝑥) =

15

𝑥+1. 𝐹𝑖𝑛𝑑 𝑒𝑎𝑐ℎ 𝑎𝑛𝑑 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦.

a) (𝑓(𝑔(𝑥)) b) (𝑔 ○ 𝑓)(𝑥) c) (𝑓(𝑓(𝑥)) d) (𝑔(𝑔(𝑥))

3. Use your answers for a) and B) in problem 2 to calculate the two problems below.

a) (𝑓(𝑔(−1)) b) (𝑔 ○ 𝑓)(3)

May 2020

May 2020

Translating My Composition

Decomposing Functions: is a process by which you can break down one complex function into multiple

smaller functions. By doing this, you can solve for functions in shorter, easier-to-understand pieces. There

may be more than one way to decompose a composite function.

Examples:

Let 𝑓(𝑥) = 𝑥 + 5, 𝑔(𝑥) = 𝑥2, ℎ(𝑥) = 3𝑥 𝑎𝑛𝑑 𝑗(𝑥) = 2𝑥. Express each function as a composite of 𝑓, 𝑔

ℎ, and/or j. 1. 𝐴(𝑥) = 𝑥4 [Think: What multiple functions composed this?]

The two functions are

g(𝑥) = 𝑥2 and g(𝑥) = 𝑥2

Check:

𝑔(𝑔(𝑥)) = 𝑔(𝑥2)

= (𝑥2)2

= 𝑥4

2. 𝐶(𝑥) = 3𝑥 + 15 [Think: What multiple functions composed this?]

The two functions are

ℎ(𝑥) = 3𝑥 and 𝑓(𝑥) = 𝑥 + 5

Check:

ℎ(𝑓(𝑥)) = ℎ(𝑥 + 5 )

= 3(𝑥 + 5)

= 3𝑥 + 15

3. 𝐶(𝑥) = 3(𝑥 + 5)2 [Think: What multiple functions composed this?]

The three functions are

ℎ(𝑥) = 3𝑥 and 𝑓(𝑥) = 𝑥 + 5 and 𝑔(𝑥) = 𝑥2,

Check:

ℎ(𝑔(𝑓(𝑥)))) = ℎ(𝑔(𝑥 + 5 ) [ work inside first- substitute 𝑥 + 5 into 𝑓(𝑥)]

= ℎ((𝑥 + 5)2) [ For every 𝑥 in 𝑔(𝑥), substitute 𝑥 + 5]

= 3(𝑥 + 5)2 [For every 𝑥 in ℎ(𝑥), substitute 𝑔(𝑓(𝑥)) which is (𝑥 + 5)2]

May 2020

May 2020

Practice Problems:

Let 𝑓(𝑥) = 𝑥2, 𝑔(𝑥) = 5𝑥, 𝑎𝑛𝑑 ℎ(𝑥) = √𝑥 + 2. Express each function as a composite of 𝑓, 𝑔 and/or ℎ. 1. F(𝑥) = 𝑥4

2. 𝐶(𝑥) = 5𝑥2

3. 𝑃(𝑥) = 𝑥 + 2

4. 𝑅(𝑥) = 5√𝑥 + 10

5. 𝑄(𝑥) = 25𝑥

7. 𝐷(𝑥) = √√𝑥 + 2 + 2

8. 𝐵(𝑥) = 𝑥 + 4√𝑥 + 4

May 2020

May 2020

Different Combinations

SET Problems: Identifying the 2 functions that make up a composite function.

Find functions 𝑓 and 𝑔 so that 𝑓 ○ 𝑔 = 𝐻

1. 𝐻(𝑥) = √𝑥2 + 5𝑥 − 4

2. 𝐻(𝑥) = (3 −1

𝑥)2

3. 𝐻(𝑥) = (3𝑥 − 7)4 4. 𝐻(𝑥) = |5𝑥2 − 78|

5. 𝐻(𝑥) = 2

3−𝑥5 6. 𝐻(𝜃) = (tan 𝜃)2

7. 𝐻(𝑥) = 9(4𝑥 − 8) + 1 8. 𝐻(𝑥) = √

1

6𝑥

May 2020

GO Problems: Finding function values given the graph. Use the graph to find all of the missing values.

1. 𝑓(■) = 8 2. 𝑓(■) = 5

3. 𝑓(■) = -1 4. 𝑓(■) = 0

5. 𝑓(−1) = 6. 𝑔(0) =

7. 𝑓(𝑥) = 𝑔(𝑥) 8. 𝑓(𝑥) − 𝑔(𝑥) = 0

9. 𝑓(𝑥) ∗ 𝑔(𝑥) = 0 10. 𝑓(2) + 𝑔(2) =

May 2020