/*! 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 14 Suppose \(h(x)=3 x^{2}-4,\) wher... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 14

Suppose \(h(x)=3 x^{2}-4,\) where the domain of \(h\) is the set of positive numbers. Find a formula for \(h^{-1}\).

Short Answer

Expert verified
The formula for \(h^{-1}\) is \(h^{-1}(x) = \sqrt{\frac{x+4}{3}}\).

Step by step solution

01

Replace function notation with variable \(y\)

We are given the function: \(h(x) = 3x^2 - 4\). Let's replace the function notation with \(y\): \[y = 3x^2 - 4\]
02

Swap \(x\) and \(y\)

Now we need to swap the \(x\) and \(y\) variables. This gives us: \[x = 3y^2 - 4\]
03

Solve for \(y\), which is \(h^{-1}(x)\)

We now need to solve for \(y\), which represents the inverse function \(h^{-1}(x)\). Start by adding 4 to both sides of the equation: \[x + 4 = 3y^2\] Next, divide both sides by 3: \[\frac{x + 4}{3} = y^2\] Now, to solve for \(y\), we need to take the square root of both sides. Since we have been given the domain of \(h(x)\) as only the set of positive numbers, we can assume that the square root of \(y^2\) will be a positive value. This gives us the inverse function \(h^{-1}(x)\): \[y = h^{-1}(x) = \sqrt{\frac{x+4}{3}}\] Thus, the formula for \(h^{-1}\) is \(h^{-1}(x) = \sqrt{\frac{x+4}{3}}\).

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

Suppose \(f\) is a function and a function \(g\) is defined by the given expression. (a) Write \(g\) as the composition of \(f\) and one or two linear functions. (b) Describe how the graph of \(g\) is obtained from the graph of \(f\). \(g(x)=-4 f(x)-7\)

Find a number \(b\) such that \(f \circ g=g \circ f,\) where \(f(x)=2 x+b\) and \(g(x)=3 x+4\).

Find a number \(c\) such that \(f \circ g=g \circ f,\) where \(f(x)=5 x-2\) and \(g(x)=c x-3\) .

Give an example to show that the product of two one-to-one functions is not necessarily a one-to-one function.

Suppose \(f\) and \(g\) are functions, each with domain of four numbers, with \(f\) and \(g\) defined by the tables below: $$\begin{array}{c|c}x & f(x) \\\\\hline 1 & 4 \\\2 & 5 \\\3 & 2 \\\4 & 3\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & 3 \\\3 & 2 \\\4 & 4 \\\5 & 1\end{array}$$ Give the table of values for \(f^{-1} \circ f\).
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.