Ch. 2 - Graphs and Functions

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

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 87b

Chapter 3, Problem 87b

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

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Identify the composition of functions (g $\circ$ f)(x), which means g(f(x)). This involves substituting the entire function f(x) into g(x).

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

Substitute $ f(x) $ into $ g(x) $ to get $ (g \circ f)(x) = g\left( \frac{1}{x-2} \right) = \frac{1}{\frac{1}{x-2}} $.

Simplify the expression $ \frac{1}{\frac{1}{x-2}} $ by multiplying numerator and denominator appropriately to find the simplified form of $ (g \circ f)(x) $.

Determine the domain of $ (g \circ f)(x) $ by considering the restrictions from both $ f(x) $ and $ g(x) $: exclude values where $ x-2=0 $ (to avoid division by zero in $ f(x) $) and where $ f(x) = 0 $ (to avoid division by zero in $ g(f(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 (g∘f)(x) = g(f(x)). It requires substituting the entire output of f(x) into g, 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 (g∘f)(x) includes all x-values in the domain of f for which f(x) is in the domain of g.

Restrictions from Rational Functions

Rational functions like f(x) = 1/(x-2) and g(x) = 1/x have restrictions where denominators cannot be zero. Identifying these values is essential to determine the domain and ensure the function is defined.

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Restrictions on Rational Equations

Related Practice

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.

$y^{3} = x + 4$

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Textbook Question

Given functions f and g, find (ƒ∘g)(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)=1/(x+4), g(x)=-(1/x)

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

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

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