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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 43a
Chapter 3, Problem 43a

Find ƒ+g and determine the domain for each function.
f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

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Step 1: Understand the problem. You are tasked with finding the sum of two functions, f(x) and g(x), which are rational functions. The sum of two functions is defined as (f + g)(x) = f(x) + g(x). Additionally, you need to determine the domain of the resulting function.
Step 2: Write the expressions for f(x) and g(x). Here, f(x) = (5x + 1) / (x² - 9) and g(x) = (4x - 2) / (x² - 9). Since both functions have the same denominator, you can add the numerators directly while keeping the common denominator.
Step 3: Add the numerators of f(x) and g(x). Combine (5x + 1) and (4x - 2) to get the new numerator: (5x + 1) + (4x - 2). Simplify this expression to get 9x - 1. The resulting function is (f + g)(x) = (9x - 1) / (x² - 9).
Step 4: Determine the domain of the resulting function. The domain of a rational function excludes any values of x that make the denominator equal to zero. For the denominator x² - 9, solve the equation x² - 9 = 0. Factorize it as (x - 3)(x + 3) = 0, which gives x = 3 and x = -3. These values are excluded from the domain.
Step 5: Write the domain in interval notation. The domain of the function is all real numbers except x = 3 and x = -3. In interval notation, this is expressed as (-∞, -3) ∪ (-3, 3) ∪ (3, ∞).

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Key Concepts

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

Function Addition

Function addition involves combining two functions, f(x) and g(x), to create a new function, ƒ+g. This is done by adding their outputs for each input x, resulting in (f+g)(x) = f(x) + g(x). Understanding this concept is crucial for solving the problem as it requires the correct application of addition to the given functions.
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Adding & Subtracting Functions Example 1

Domain of a Function

The domain of a function is the set of all possible input values (x) for which the function is defined. For rational functions like f(x) and g(x), the domain is restricted by values that make the denominator zero. Identifying these restrictions is essential to determine the valid inputs for the combined function ƒ+g.
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Domain Restrictions of Composed Functions

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In this case, both f(x) and g(x) are rational functions with a common denominator (x² - 9). Understanding the properties of rational functions, including their behavior near vertical asymptotes and discontinuities, is important for analyzing the resulting function and its domain.
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Related Practice
Textbook Question

Give the slope and y-intercept of each line whose equation is given. Then graph the linear function. f(x) = (3/4)x-2

Textbook Question

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = (x − 1)², x ≤ 1

Textbook Question

Use the graph of y = f(x) to graph each function g. g(x) = (1/2)f(2x)

Textbook Question

Find f−g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

Textbook Question

Find fg and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

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Textbook Question

In Exercises 41–44, use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (4, -7) and parallel to the line whose equation is 3x + y - 9 = 0.