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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 19b
Chapter 3, Problem 19b

For the pair of functions defined, find (ƒ-g)(x).Give the domain of each. See Example 2.
ƒ(x)=3x+4, g(x)=2x-6

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Identify the given functions: \( f(x) = 3x + 4 \) and \( g(x) = 2x - 6 \).
Find the difference of the functions, which is defined as \( (f - g)(x) = f(x) - g(x) \).
Substitute the expressions for \( f(x) \) and \( g(x) \) into the difference: \( (f - g)(x) = (3x + 4) - (2x - 6) \).
Simplify the expression by distributing the negative sign and combining like terms: \( (f - g)(x) = 3x + 4 - 2x + 6 \).
Determine the domain of each function. Since both \( f(x) = 3x + 4 \) and \( g(x) = 2x - 6 \) are linear functions, their domains are all real numbers, \( (-\infty, \infty) \). The domain of \( (f - g)(x) \) is also all real numbers.

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Key Concepts

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

Function Operations (Addition and Subtraction)

Function operations involve combining two functions by adding or subtracting their outputs for the same input value. For (ƒ - g)(x), subtract g(x) from ƒ(x) to create a new function. This helps analyze how two functions interact algebraically.
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Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When combining functions, the domain of the resulting function is the intersection of the domains of the original functions, ensuring all operations are valid.
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Linear Functions

Linear functions have the form f(x) = mx + b, where m and b are constants. They produce straight-line graphs and are defined for all real numbers. Understanding their properties simplifies finding combined functions and their domains.
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Related Practice
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Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. through (-1,3), and (3,4)

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For the pair of functions defined, find (f/g)(x).Give the domain of each. See Example 2.

ƒ(x)=3x+4, g(x)=2x-8

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For each graph, determine whether y is a function of x. Give the domain and range of each relation.

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For the pair of functions defined, find (ƒg)(x).Give the domain of each. See Example 2.

ƒ(x)=3x+4, g(x)=2x-7

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Textbook Question

For each piecewise-defined function, find (a) ƒ(-5), (b) ƒ(-1), (c) ƒ(0), and (d) ƒ(3).

f(x)={2+xif x<−4−xif −4≤x≤23xif x>2f(x) =\(\begin{cases}\)2 + x & \(\text{if }\) x < -4 \\-x & \(\text{if }\) -4 \(\leq\) x \(\leq\) 2 \\3x & \(\text{if }\) x > 2\(\end{cases}\)

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Textbook Question

For the pair of functions defined, find (ƒ+g)(x).Give the domain of each. See Example 2.

ƒ(x)=3x+4, g(x)=2x-5

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