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Chapter 1: Problem 44

Find formulas for \(f \circ g\) and \(g \circ f,\) and state the domains of the functions. $$f(x)=\frac{x}{1+x^{2}}, g(x)=\frac{1}{x}$$

Short Answer

Expert verified
For \( f \circ g \), the formula is \( \frac{x}{x^2+1} \) with domain \( x \neq 0 \). For \( g \circ f \), the formula is \( \frac{1+x^2}{x} \) with domain \( x \neq 0 \).

Step by step solution

01

Understand the Functions

We have two functions: \( f(x) = \frac{x}{1+x^2} \) and \( g(x) = \frac{1}{x} \). We need to determine the composition of these functions in two ways: \( f \circ g \) and \( g \circ f \).
02

Find \( f \circ g \)

To find \( f \circ g \), we substitute \( g(x) \) into \( f(x) \). Thus, the composition \ \( f(g(x)) = f\left(\frac{1}{x}\right) = \frac{\frac{1}{x}}{1+\left(\frac{1}{x}\right)^2} = \frac{1/x}{1+1/x^2} \ = \frac{1}{x} \left/ \left( \frac{x^2 + 1}{x^2} \right) \right. = \frac{1}{x} \cdot \frac{x^2}{x^2 + 1} = \frac{x^2}{x(x^2+1)} = \frac{x}{x^2+1} \).
03

Determine the Domain of \( f \circ g \)

The domain of \( f(g(x)) \) is determined by the domain of \( g(x) \), which is \( x eq 0 \), and the resulting expression \( \frac{x}{x^2+1} \), which is defined for all real numbers. Therefore, the domain of \( f \circ g \) is all real numbers except 0, i.e., \( x \in \mathbb{R} \setminus \{0\} \).
04

Find \( g \circ f \)

To find \( g \circ f \), we substitute \( f(x) \) into \( g(x) \). Thus, the composition \ \( g(f(x)) = g\left(\frac{x}{1+x^2}\right) = \frac{1}{\frac{x}{1+x^2}} \ = \frac{1+x^2}{x} \).
05

Determine the Domain of \( g \circ f \)

The domain of \( g(f(x)) \) is determined by both \( f(x) \)'s domain (all real numbers; no restrictions), and the resulting expression \( \frac{1+x^2}{x} \). The main restriction comes from \( x \) cannot be 0, so the domain of \( g \circ f \) is \( x \in \mathbb{R} \setminus \{0\} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain of a Function
Understanding the domain of a function is fundamental when dealing with functions and their compositions. The domain refers to all permissible input values (or x-values) that can be fed into a function without causing mathematical issues like division by zero or involving square roots of negative numbers.

For example, when dealing with the function \( g(x) = \frac{1}{x} \), we immediately see a restriction: division by zero is undefined. Therefore, \( g(x) \) cannot accept \( x = 0 \).
  • For \( f(x) = \frac{x}{1+x^2} \), the function is defined for all real numbers as the denominator never becomes zero.
  • In the composition \( f \circ g \), we substitute \( g(x) \) into \( f(x) \), and thus must respect \( g's \) domain restriction. So, \( x eq 0 \).
  • Similarly, for \( g \circ f \), knowing \( f(x) \) is always defined, the domain restriction still comes from the expression \( \frac{1+x^2}{x} \), again requiring \( x eq 0 \).
Function Operations
Function operations include addition, subtraction, multiplication, division, and composition. These operations allow us to combine two functions into a single new function.

When considering function operations, attention to domains is crucial. Each type of operation may pose additional restrictions. For example:
  • Addition and subtraction might have domains restricted to the intersection of the two function domains.
  • Multiplication and division often impose specific domain restrictions, particularly division, which cannot allow division by zero.
  • Composition, a type of operation where functions input other functions, often combines these restrictions in a unique way that must be carefully calculated.
Understanding these restrictions helps avoid undefined results and ensures the overall integrity of the newly formed function.
Composite Functions
Composite functions involve creating a new function by applying one function to the results of another. The notation \( f \circ g \) means "f composed with g," and it is calculated by substituting \( g(x) \) into \( f(x) \).

To correctly find the composite functions \( f \circ g \) and \( g \circ f \), we follow these steps:
  • Calculate \( f(g(x)) \): Substitute the entire expression \( g(x) \) into \( f(x) \).
  • Calculate \( g(f(x)) \): Substitute \( f(x) \) into \( g(x) \).
  • After substituting, simplify the expression wherever possible to determine the simplest form.
Evaluating composite functions not only requires these substitutions but also an analysis of the resulting expression's domain, which is crucial for understanding what inputs are suitable. This ensures that the new function provides meaningful and correct output values, honoring all mathematical rules and restrictions.

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Most popular questions from this chapter

Sketch the graph of the equation by trans lating. reflecting. compressing. and stretching the graph of \(y=|x|\) appropriately. and then use a graphing utility to confirm that your sketch is correct. $$y=1-|x-3|$$

Sketch the graph of \(y=x+(1 / x)\) by adding corresponding y-coordinates on the graphs of \(y=x\) and \(y=1 / x .\) Use a graphing utility to confirm that your sketch is correct.

The spring in a heavy-duty shock absorber has a natural length of \(3 \mathrm{ft}\) and is compressed \(0.2 \mathrm{ft}\) by a load of 1 ton. \(\mathrm{An}\) additional load of 5 tons compresses the spring an additional \(1 \mathrm{ft}\). (a) Assuming that Hooke's law applies to compression as well as extension, find an equation that expresses the length \(y\) that the spring is compressed from its natural length (in feet) in terms of the load \(x\) (in tons). (b) Graph the equation obtained in part (a). (c) Find the amount that the spring is compressed from its natural length by a load of 3 tons. (d) Find the maximum load that can be applied if safety regulations prohibit compressing the spring to less than half its natural length.

Based on your knowledge of the graphs of \(y=x\) and \(y=\sin x,\) make a sketch of the graph of \(y=x \sin x .\) Check your conclusion using a graphing utility.

Express \(f\) as a composition of two functions; that is, find \(g\) and \(h\) such that \(f=g \circ h .\) [Note: Each exercise has more than one solution. \(]\) (a) \(f(x)=3 \sin \left(x^{2}\right)\) (b) \(f(x)=3 \sin ^{2} x+4 \sin x\)
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