/*! 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 54 Let \(g(x)=x^{2}+3 .\) Find a fu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 54

Let \(g(x)=x^{2}+3 .\) Find a function \(f\) that produces the given composition. $$(g \circ f)(x)=x^{2 / 3}+3$$

Short Answer

Expert verified
Answer: The function \(f(x) = x^{\frac{1}{3}}\) produces the given composition \((g \circ f)(x) = x^{\frac{2}{3}} + 3\) with \(g(x) = x^2 + 3\).

Step by step solution

01

Express the composition as \(g(f(x))\)

Given: \((g \circ f)(x)=x^{\frac{2}{3}}+3\). This means \(g(f(x)) = x^{\frac{2}{3}} + 3\).
02

Solve for \(f(x)\) in terms of g

We know that \(g(x) = x^2 + 3\). Let's say \(f(x)=a\). So we have \(g(a) = a^2 + 3 = x^{\frac{2}{3}} + 3\).
03

Isolate \(a^2\)

We want to extract \(a\) from this equation to find \(f(x)\). So, we rewrite the equation and subtract \(3\) from both sides: $$a^2 = x^{\frac{2}{3}}$$
04

Solve for \(a\)

Since we have \(a^2\), to find \(a\) (which is our \(f(x)\)) we just need to take the square root of both sides of the equation: $$\sqrt{a^2} = \sqrt{x^{\frac{2}{3}}}$$ Now, we have: $$a = x^{\frac{1}{3}}$$
05

Write the final \(f(x)\)

The solution is the function \(f(x) = x^{\frac{1}{3}}\). This function will produce the given composition \((g \circ f)(x)\) when plugged in place of \(x\) in function \(g(x)\): $$(g \circ f)(x) = g\left(x^{\frac{1}{3}}\right) = \left(x^{\frac{1}{3}}\right)^2 + 3 = x^{\frac{2}{3}} + 3$$

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

Imagine a lidless box with height \(h\) and a square base whose sides have length \(x\). The box must have a volume of \(125 \mathrm{ft}^{3}\) a. Find and graph the function \(S(x)\) that gives the surface area of the box, for all values of \(x>0\) b. Based on your graph in part (a), estimate the value of \(x\) that produces the box with a minimum surface area.

Sketch a graph of the given pair of functions to conjecture a relationship between the two functions. Then verify the conjecture. $$\tan ^{-1} x ; \frac{\pi}{2}-\cot ^{-1} x$$

The velocity of a skydiver (in \(\mathrm{m} / \mathrm{s}\) ) \(t\) seconds after jumping from the plane is \(v(t)=600\left(1-e^{-k t / 60}\right) / k\), where \(k > 0\) is a constant. The terminal velocity of the skydiver is the value that \(v(t)\) approaches as \(t\) becomes large. Graph \(v\) with \(k=11\) and estimate the terminal velocity.

Let \(f\) be an \(n\) th-degree polynomial and let \(g\) be an \(m\) th-degree polynomial. What is the degree of the following polynomials? a. \(f \cdot f\) b. \(f^{\circ} f\) c. \(f \cdot g\) d. \(f^{\circ} g\)

Identify the amplitude and period of the following functions. $$q(x)=3.6 \cos (\pi x / 24)$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

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.