Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 87a

Chapter 3, Problem 87a

Given functions f and g, find (ƒ∘g)(x) and its domain. ƒ(x)=1/(x-2), g(x)=1/x

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Identify the given functions: $f(x) = \frac{1}{x-2}$ and $g(x) = \frac{1}{x}$.

Find the composition $(f \circ g)(x)$, which means $f(g(x))$. Substitute $g(x)$ into $f$: $f(g(x)) = f\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x} - 2}$.

Simplify the expression inside the denominator: $\frac{1}{\frac{1}{x} - 2} = \frac{1}{\frac{1 - 2x}{x}}$.

Rewrite the complex fraction by multiplying numerator and denominator appropriately: $\frac{1}{\frac{1 - 2x}{x}} = \frac{1 \cdot x}{1 - 2x} = \frac{x}{1 - 2x}$.

Determine the domain of $(f \circ g)(x)$ by considering the domains of $f$ and $g$ and restrictions from the composition: exclude values where $g(x)$ is undefined (i.e., $x \neq 0$) and where the denominator in $f(g(x))$ is zero (i.e., $1 - 2x \neq 0$).

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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 (f∘g)(x) = f(g(x)). It requires substituting the entire output of g(x) into the function f, creating a new function that combines both operations.

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. When composing functions, the domain of (f∘g)(x) includes all x-values in the domain of g for which g(x) is in the domain of f.

Rational Functions and Restrictions

Rational functions are ratios of polynomials and often have restrictions where the denominator equals zero. Identifying these restrictions is crucial to determine the domain, especially when composing functions that involve division.

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Restrictions on Rational Equations

Related Practice

Textbook Question

Determine the largest open intervals of the domain over which each function is (c) constant. See Example 9.

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Textbook Question

Determine the largest open intervals of the domain over which each function is b) decreasing. See Example 9.

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Textbook Question

Given functions f and g, (g∘ƒ)(x) and its domain. ƒ(x)=1/(x-2), g(x)=1/x

Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=√x, g(x)=3/(x+6)

Textbook Question

Determine the largest open intervals of the domain over which each function is (a) increasing. See Example 9.

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Textbook Question

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. $5 y^{2} + 5 x^{2} = 30$

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