Algebra A Combined Function

Elayn Martin-Gay

ISBN #9780321726391

4th Edition

6,670 Questions

28,872 Students Helped

Homework Questions

Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

This section explains the algebra of functions, which includes adding, subtracting, multiplying, and dividing functions to create new functions, as well as composing functions where the output of one serves as the input of another. Key applications, such as converting Fahrenheit to Kelvins, show the practical utility of these operations. Remembering domain restrictions and performing operations carefully ensures accurate results.

Learning Objectives

Understand how to add, subtract, multiply, and divide functions, and recognize that the results form new functions.

Master the concept of the algebra of functions including the operations of function arithmetic.

Learn to compose functions, using the output of one function as the input of another.

Apply these techniques to real-world problems such as converting temperature scales using function composition.

Key Concepts

CONCEPT

DEFINITION

Algebra of Functions

The process of creating new functions by performing operations such as addition, subtraction, multiplication, and division on existing functions.

Function Composition

The operation of applying one function to the results of another, expressed as (g ∘ f)(x) = g(f(x)).

Domain Restriction

The rule that when dividing two functions, the denominator function cannot equal zero, ensuring the resulting composite function is well-defined.

Example Problems

Example 1

For the functions $f$ and $g$, find a. $(f+g)(x)$, b. $(f-g)(x), c .(f \cdot g)(x)$, and d. $\left(\frac{f}{g}\right)(x)$. $$ f(x)=x-7 ; g(x)=2 x+1 $$

Example 2

For the functions $f$ and $g$, find a. $(f+g)(x)$, b. $(f-g)(x), c .(f \cdot g)(x)$, and d. $\left(\frac{f}{g}\right)(x)$. $$ f(x)=x+4 ; g(x)=5 x-2 $$

Example 3

For the functions $f$ and $g$, find a. $(f+g)(x)$, b. $(f-g)(x), c .(f \cdot g)(x)$, and d. $\left(\frac{f}{g}\right)(x)$. $$ f(x)=x^{2}+1 ; g(x)=5 x $$

Example 4

For the functions $f$ and $g$, find a. $(f+g)(x)$, b. $(f-g)(x), c .(f \cdot g)(x)$, and d. $\left(\frac{f}{g}\right)(x)$. $$ f(x)=x^{2}-2 ; g(x)=3 x $$

Example 5

For the functions $f$ and $g$, find a. $(f+g)(x)$, b. $(f-g)(x), c .(f \cdot g)(x)$, and d. $\left(\frac{f}{g}\right)(x)$. $$ f(x)=\sqrt{x} ; g(x)=x+5 $$

Step-by-Step Explanations

QUESTION

Given f(x) = x + 3 and g(x) = 3x - 1, find the function (f + g)(x).

STEP-BY-STEP ANSWER:

Step 1: Write down the definitions: f(x) = x + 3 and g(x) = 3x - 1.
Step 2: Apply the definition of the sum of functions: (f + g)(x) = f(x) + g(x).
Step 3: Substitute the expressions: (f + g)(x) = (x + 3) + (3x - 1).
Step 4: Combine like terms: x + 3x = 4x and 3 - 1 = 2, so (f + g)(x) = 4x + 2.
Final Answer: (f + g)(x) = 4x + 2.

Sum of Functions

QUESTION

Given f(x) = (5/9)(x - 32) and g(x) = x + 273.15, find (g ∘ f)(x) to convert Fahrenheit directly to Kelvins.

STEP-BY-STEP ANSWER:

Step 1: Identify the inner function and the outer function: f converts Fahrenheit to Celsius and g converts Celsius to Kelvins.
Step 2: Write the composition: (g ∘ f)(x) = g(f(x)).
Step 3: Substitute f(x) into g: (g ∘ f)(x) = f(x) + 273.15, where f(x) = (5/9)(x - 32).
Step 4: Thus, the composed function is (g ∘ f)(x) = (5/9)(x - 32) + 273.15.
Final Answer: (g ∘ f)(x) = (5/9)(x - 32) + 273.15.

Function Composition

Common Mistakes

Mistaking the arithmetic operations on functions where the operation must be applied to the entire function, not the individual terms beforehand.
Forgetting to simplify and combine like terms after performing function operations.
Overlooking the domain restrictions, especially when dividing functions (omitting the condition g(x) ? 0).
Confusing function composition with simple multiplication or addition of functions.