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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 2 - Functions and GraphsProblema 45b
Capitolo 3, Problema 45b

Find f−g and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

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Step 1: Understand the problem. You are tasked with finding the difference of two functions, f(x) and g(x), which is expressed as (f - g)(x) = f(x) - g(x). Additionally, you need to determine the domain of the resulting function. The given functions are f(x) = 8x / (x - 2) and g(x) = 6 / (x + 3).
Step 2: Write the expression for (f - g)(x). Subtract g(x) from f(x): (f - g)(x) = (8x / (x - 2)) - (6 / (x + 3)). To subtract these fractions, you need a common denominator.
Step 3: Find the common denominator. The denominators of the two fractions are (x - 2) and (x + 3). The least common denominator (LCD) is the product of these two denominators: (x - 2)(x + 3). Rewrite each fraction with this common denominator.
Step 4: Rewrite the fractions with the common denominator. For the first fraction, multiply both numerator and denominator by (x + 3): (8x / (x - 2)) becomes (8x(x + 3) / ((x - 2)(x + 3))). For the second fraction, multiply both numerator and denominator by (x - 2): (6 / (x + 3)) becomes (6(x - 2) / ((x - 2)(x + 3))).
Step 5: Combine the fractions. Now that both fractions have the same denominator, subtract the numerators: (f - g)(x) = [(8x(x + 3)) - (6(x - 2))] / [(x - 2)(x + 3)]. Simplify the numerator by distributing and combining like terms. Finally, determine the domain by identifying the x-values that make the denominator zero. These are x = 2 and x = -3, which must be excluded from the domain.

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Function Subtraction

Function subtraction involves finding the difference between two functions, denoted as (f - g)(x) = f(x) - g(x). In this case, you will subtract the output of g(x) from f(x) for each value of x in the domain where both functions are defined.
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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 like f(x) and g(x), the domain excludes values that make the denominator zero, as these would result in undefined outputs.
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Finding Common Denominators

When subtracting two rational functions, it is often necessary to find a common denominator to combine the fractions. This involves identifying the least common multiple of the denominators, which allows for the proper subtraction of the two functions while maintaining equivalent values.
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