Section 5.7 : Combining Functions

Learning Targets: F.BF.1.b

Important Terms and Definitions

We are able to take all of the knowledge we have about arithmetic operations and apply it to functions. Functions can be added, subtracted, multiplied, and divided.

Addition: (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥)

Subtraction: (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)

Multiplication: (𝑓 ∙ 𝑔)(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥)

Division: (𝑓𝑔

) (𝑥) = 𝑓(𝑥)𝑔(𝑥)

, 𝑔(𝑥) ≠ 0

Example: Given that 𝑓(𝑥) = 3𝑥 + 4 and 𝑔(𝑥) = 2𝑥 − 3, find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓 ∙ 𝑔)(𝑥), and (𝑓

𝑔) (𝑥).

(𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥) = 3𝑥 + 4 + 2𝑥 − 3 = 5𝑥 + 1

(𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥) = 3𝑥 + 4 − (2𝑥 − 3) = 3𝑥 + 4 − 2𝑥 + 3 = 𝑥 + 7

(𝑓 ∙ 𝑔)(𝑥) = 𝑓(𝑥) ∙ 𝑔(𝑥) = (3𝑥 + 4)(2𝑥 − 3) = 6𝑥2 − 9𝑥 + 6𝑥 − 12

= 6𝑥2 − 3𝑥 − 12

(𝑓𝑔

) (𝑥) = 𝑓(𝑥)𝑔(𝑥)

= 3𝑥+42𝑥−3

(ex 1) Given that 𝑓(𝑥) = 6𝑥 − 1 and 𝑔(𝑥) = 3𝑥, find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓 ∙ 𝑔)(𝑥), and (𝑓

𝑔) (𝑥).

(ex 2) Given that 𝑓(𝑥) = 144𝑥 − 48 and 𝑔(𝑥) = 12, find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓 ∙ 𝑔)(𝑥), and (𝑓

𝑔) (𝑥).

Finding a Combination of Functions at a Given Value

Example: Given that 𝑓(𝑥) = 3𝑥 + 4 and 𝑔(𝑥) = 2𝑥 − 3, find (𝑓 + 𝑔)(3).

(𝑓 + 𝑔)(3) = 𝑓(3) + 𝑔(3) = 3(3) + 4 + 2(3) − 3 = 9 + 4 + 6 − 3 = 16

(ex 3) Given that 𝑓(𝑥) = 3 ∙ 8𝑥 and 𝑔(𝑥) = 2 ∙ 8𝑥 + 1, find (𝑓 + 𝑔)(2) and (𝑓 − 𝑔)(3).

(ex 4) Given that 𝑓(𝑥) = 5 ∙ 3𝑥 and 𝑔(𝑥) = 2 ∙ 3𝑥 + 6, find (𝑓 ∙ 𝑔)(2) and (𝑓𝑔

) (3).

(ex 5) Scientists conducted a study of the sparrow populations in Boardman and Canfield. The results of the study projected that the population in Boardman in x years will be modeled by 𝐵(𝑥) = 950 ∙ 1.29𝑥 − 120𝑥 and the population in Canfield in x years will be modeled by 𝐶(𝑥) = 500𝑥 − 600. What is a function giving the difference in population of sparrows between Boardman and Canfield? Predict what the difference in population will be in 6 years.

Homework – Section 5.7 : Combining Functions

Find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓 ∙ 𝑔)(𝑥), and (𝑓𝑔) (𝑥) for the following pairs of functions.

1. 𝑓(𝑥) = 15𝑥 + 3, 𝑔(𝑥) = 3𝑥 2. 𝑓(𝑥) = 2𝑥, 𝑔(𝑥) = 5𝑥 + 2 3. 𝑓(𝑥) = 3𝑥 + 2, 𝑔(𝑥) = 5𝑥 + 1

4. Given 𝑓(𝑥) = 9 ∙ 2𝑥 + 16𝑥 − 6, 𝑔(𝑥) = (−8) ∙ 2𝑥 − 17, find (𝑓 + 𝑔)(𝑥), (𝑓 + 𝑔)(4), (𝑓 − 𝑔)(𝑥) and (𝑓 − 𝑔)(1).

5. Given 𝑓(𝑥) = 9𝑥 + 3 − 6𝑥, 𝑔(𝑥) = 3𝑥, find (𝑓 ∙ 𝑔)(𝑥), (𝑓 ∙ 𝑔)(3), (𝑓𝑔) (𝑥) and

(𝑓𝑔) (4).

Write a pair of functions that have the following characteristics.

6. (𝑓 − 𝑔)(𝑥) = 2𝑥 + 3 7. (𝑓 ∙ 𝑔)(𝑥) = 12𝑥 + 8 8. (𝑓 + 𝑔)(𝑥) = 4 ∙ 6𝑥 − 2

9. The cost of producing x baseball gloves is modeled by 𝐶(𝑥) = 20 + 15𝑥. The markup price of selling baseball gloves is modeled by 𝑀(𝑥) = 1.5. What is a function giving the selling price of the gloves? How much would a coach spend if he wanted to buy gloves for 9 of his players?

10. The number of widgets produced in x days will be modeled by the function 𝑊(𝑥) = 800 + 16𝑥 and the number of defective widgets in x days will be modeled by 𝐷(𝑥) = 4 + 2𝑥. Write a function giving the percent of defective widgets. Predict the percent of defective widgets produced in 12 days. Round to the nearest tenth of a percent.