IntroductionFunctions are relations in which each element in the domain is mapped to exactly one element in the range; that is, for every value of x, there is exactly one value of y. In order to understand how functions are related, we need to understand how to perform different arithmetic operations on functions. In this lesson, we will add, subtract, multiply, and divide exponential and linear functions.

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3.7.1: Operating on Functions

Key Concepts• Combine linear and exponential expressions using

addition: (f + g)(x) = f(x) + g(x). In other words, add the two functions together by combining like terms.

• Combine linear and exponential expressions using subtraction: (f – g)(x) = f(x) – g(x). In other words, subtract the second function from the first while making sure to distribute the negative across all terms of the second function.

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3.7.1: Operating on Functions

Key Concepts, continued• Combine linear and exponential expressions using

multiplication: (f • g)(x) = f(x) • g(x). In other words, multiply the two functions together.

• Combine linear and exponential expressions using division: (f ÷ g)(x) = f(x) ÷ g(x). In other words, divide the first function by the second function. Use a fraction bar to display the final function.

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3.7.1: Operating on Functions

Common Errors/Misconceptions• incorrectly using the distributive property when

subtracting two expressions

• incorrectly using the order of operations when multiplying or dividing two expressions

• incorrectly using the distributive property when multiplying two expressions

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3.7.1: Operating on Functions

Guided Practice

Example 1If f(x) = 3x + 2 and g(x) = 2x – 7, what is the result of adding the two functions? What is (f + g)(x)? How do you represent this algebraically?

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3.7.1: Operating on Functions

Guided Practice: Example 1, continued

1. Add the two function rules.

(f + g)(x) = f(x) + g(x)

Since f(x) = 3x + 2 and g(x) = 2x – 7, (f + g)(x) = (3x + 2) + (2x – 7).

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3.7.1: Operating on Functions

Guided Practice: Example 1, continued

2. Combine like terms.Clear the parentheses and reorder the terms on the right side of the equation.

(f + g)(x) = (3x + 2) + (2x – 7) Equation

(f + g)(x) = 3x + 2 + 2x – 7 Remove the parentheses.

(f + g)(x) = 3x + 2x + 2 – 7 Reorder the terms: variables with coefficients first, followed by constants.

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3.7.1: Operating on Functions

Guided Practice: Example 1, continued

3. Simplify the equation.

(f + g)(x) = 3x + 2x + 2 – 7 Equation

(f + g)(x) = 5x – 5

The result of adding  f(x) = 3x + 2 and g(x) = 2x – 7 is (f + g)(x) = 5x – 5.

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3.7.1: Operating on Functions

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3.7.1: Operating on Functions

Guided Practice: Example 1, continued

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Guided Practice

Example 3If f(x) = 2x – 3 and g(x) = 4x – 11, what is the result of subtracting the two functions? What is (f – g)(x)? How do you represent this algebraically?

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3.7.1: Operating on Functions

Guided Practice: Example 3, continued

1. Subtract the two function rules.(f – g)(x) = f(x) – g(x)

Since f(x) = 2x – 3 and g(x) = 4x – 11, (f – g)(x) =

(2x – 3) – (4x – 11).

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3.7.1: Operating on Functions

Guided Practice: Example 3, continued

2. Combine like terms.

Clear the parentheses and reorder the terms on the right side of the equation. Be careful to correctly distribute the negative sign.

(f – g)(x) = (2x – 3) – (4x – 11) Equation

(f – g)(x) = 2x – 3 – 4x + 11 Distribute the negative.

(f – g)(x) = 2x – 4x – 3 + 11 Reorder the terms:

variables with coefficients first,

followed by constants. 12

3.7.1: Operating on Functions

Guided Practice: Example 3, continued

3. Simplify the equation.(f – g)(x) = 2x – 4x – 3 + 11 Equation

(f – g)(x) = –2x + 8

(f – g)(x) = –2x + 8 is the result of subtracting  f(x) = 2x – 3 and g(x) = 4x – 11.

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3.7.1: Operating on Functions

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3.7.1: Operating on Functions

Guided Practice: Example 3, continued