/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 47 Let \(f(x)=|x|, g(x)=x^{2}-4\) \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 47

Let \(f(x)=|x|, g(x)=x^{2}-4\) \(F(x)=\sqrt{x},\) and \(G(x)=1 /(x-2) .\) Determine the following composite functions and give their domains. $$g \circ g$$

Short Answer

Expert verified
Answer: The composite function \(g\circ g\) for the function \(g(x) = x^2 - 4\) is \((x^2-4)^2-4\), and its domain is all real numbers, represented as \((-鈭�, 鈭�)\).

Step by step solution

01

Finding the composition g(g(x))

To find \(g\circ g\), we substitute \(g(x) = x^2 - 4\) into itself. Let's do this: $$g(g(x)) = g(x^2 - 4).$$ Now, substitute \(x^2-4\) into the original function \(g(x) = x^2 - 4\): $$g(g(x)) = (x^2 - 4)^2 - 4.$$
02

Determining the domain of g(g(x))

The domain of a function is the set of all possible input values (x-values) for which the function is defined. We need to find the domain of the composite function \(g(g(x)) = (x^2 - 4)^2 - 4\). Since this is a polynomial function (a quadratic function in this case), it is defined for all real numbers. Therefore, the domain of \(g(g(x)) = (x^2 - 4)^2 - 4\) is: $$\text{Domain of } g(g(x)) = (-\infty, \infty).$$ This means that the composite function \(g\circ g\) is defined for all real numbers. So, the function \(g\circ g\) and its domain are given as: $$g\circ g: (x^2-4)^2-4$$ $$\text{Domain of } g(g(x)) = (-\infty, \infty).$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Without using a calculator, evaluate or simplify the following expressions. $$\tan ^{-1}(\tan \pi / 4)$$

Find a formula for a function describing the given situation. Graph the function and give a domain that makes sense for the problem. Recall that with constant speed. distance \(=\) speed \(\cdot\) time elapsed or \(d=v t\) A function \(y=f(x)\) such that if you ride a bike for \(50 \mathrm{mi}\) at \(x\) miles per hour, you arrive at your destination in \(y\) hours

Using words and figures, explain why the range of \(f(x)=x^{n},\) where \(n\) is a positive odd integer, is all real numbers. Explain why the range of \(g(x)=x^{n},\) where \(n\) is a positive even integer, is all nonnegative real numbers.

Beginning with the graphs of \(y=\sin x\) or \(y=\cos x,\) use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility only to check your work. $$q(x)=3.6 \cos (\pi x / 24)+2$$

Use analytical methods to find the following points of intersection. Use a graphing utility only to check your work. Find the point(s) of intersection of the parabola \(y=x^{2}+2\) and the line \(y=x+4\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.