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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 46a
Chapter 3, Problem 46a

Find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x)= = 9x/(x - 4), g(x) = 7/(x+8)

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Step 1: Understand the problem. You are tasked with finding the sum (ƒ+g), difference (ƒ−g), product (ƒg), and quotient (ƒ/g) of the given functions f(x) = 9x/(x - 4) and g(x) = 7/(x + 8). Additionally, you need to determine the domain for each resulting function.
Step 2: To find ƒ+g, add the two functions: ƒ+g = (9x/(x - 4)) + (7/(x + 8)). Combine the fractions by finding a common denominator, which is (x - 4)(x + 8). Rewrite each fraction with the common denominator and simplify.
Step 3: To find ƒ−g, subtract the two functions: ƒ−g = (9x/(x - 4)) − (7/(x + 8)). Similar to addition, find a common denominator (x - 4)(x + 8), rewrite the fractions, and simplify.
Step 4: To find ƒg, multiply the two functions: ƒg = (9x/(x - 4)) * (7/(x + 8)). Multiply the numerators and denominators directly: (9x * 7)/((x - 4)(x + 8)). Simplify the expression.
Step 5: To find ƒ/g, divide the two functions: ƒ/g = (9x/(x - 4)) ÷ (7/(x + 8)). Division of fractions involves multiplying the first fraction by the reciprocal of the second: (9x/(x - 4)) * ((x + 8)/7). Simplify the resulting expression. For the domain of each function, exclude values of x that make any denominator zero (x = 4, x = -8).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Operations

Function operations involve combining two functions through addition, subtraction, multiplication, and division. For example, if f(x) and g(x) are two functions, then f+g means adding their outputs, while f-g means subtracting the output of g from f. Understanding these operations is essential for manipulating and analyzing functions in algebra.
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Multiplying & Dividing Functions

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by values that make the denominator zero, as these would lead to undefined outputs. Identifying the domain is crucial for ensuring that the operations performed on functions yield valid results.
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Domain Restrictions of Composed Functions

Rational Functions

Rational functions are ratios of two polynomial functions. They can exhibit unique behaviors, such as vertical asymptotes where the denominator is zero, and horizontal asymptotes that describe end behavior. Understanding the characteristics of rational functions is important for analyzing their graphs and determining their domains.
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Related Practice
Textbook Question

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x + 2)² + (y + 2)² = 4

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Textbook Question

Find ƒ+g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

Textbook Question

In Exercises 46–49, give the slope and y-intercept of each line whose equation is given. Then graph the line. y = (2/5)x - 1

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Textbook Question

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x)= |x|, g(x) = |x| +1

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Textbook Question

Use the graph of y = f(x) to graph each function g. g(x) = -f(x-1) + 1