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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 23c
Chapter 3, Problem 23c

For the pair of functions defined, find (ƒg)(x). Give the domain of each. See Example 2.
ƒ(x)=√(4x-1), g(x)=1/x

Verified step by step guidance
1
Identify the composition of functions (ƒg)(x), which means ƒ(g(x)). This involves substituting g(x) into the function ƒ(x).
Write the expression for (ƒg)(x) by replacing x in ƒ(x) with g(x). Since ƒ(x) = \(\sqrt{4x - 1}\) and g(x) = \(\frac{1}{x}\), we have (ƒg)(x) = \(\sqrt{4 \cdot \frac{1}{x}\) - 1}.
Simplify the expression inside the square root: \(\sqrt{\frac{4}{x}\) - 1}.
Determine the domain of g(x) = \(\frac{1}{x}\). Since division by zero is undefined, the domain of g(x) is all real numbers except x = 0.
Determine the domain of (ƒg)(x) by considering the expression inside the square root must be greater than or equal to zero: \(\frac{4}{x}\) - 1 \(\geq\) 0. Solve this inequality along with the domain restriction from g(x) to find the domain of (ƒg)(x).

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Key Concepts

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

Function Composition

Function composition involves applying one function to the result of another, denoted as (ƒg)(x) = ƒ(g(x)). It requires substituting the entire output of g(x) into the function ƒ. Understanding this process is essential to correctly combine the two given functions.
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Function Composition

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When composing functions, the domain of the composite function depends on the domains of both functions and the values for which the inner function's output fits the outer function's domain.
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Domain Restrictions of Composed Functions

Square Root and Rational Function Domains

For ƒ(x) = √(4x - 1), the expression inside the square root must be non-negative, so 4x - 1 ≥ 0. For g(x) = 1/x, x cannot be zero because division by zero is undefined. These restrictions are crucial when determining the domains of each function and their composition.
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Related Practice
Textbook Question

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

ƒ(x)=√(4x-1), g(x)=1/x

Textbook Question

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. vertical, through (-6, 4)

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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)=√(4x-1), g(x)=1/x

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Textbook Question

Determine whether the three points are the vertices of a right triangle. See Example 3.

(−2,−8),(0,−4),(−4,−7)(-2,-8),(0,-4),(-4,-7)

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Textbook Question

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

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Textbook Question

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

ƒ(x)=√(4x-1), g(x)=1/x

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