/*! 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 548 Find the inverse of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 548

Find the inverse of the function on the given domain. \(f(x)=(x+2)^{2}, \quad[-2, \infty)\)

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = \sqrt{x} - 2 \) for \( x \geq 0 \).

Step by step solution

01

Understand the Problem

We are given a function \( f(x) = (x+2)^2 \) on the domain \([-2, \infty)\). We need to find its inverse function.
02

Switch x and y

Start by rewriting the function as an equation with \(y\): \( y = (x + 2)^2 \). To find the inverse, swap \(x\) and \(y\) to get: \( x = (y + 2)^2 \).
03

Solve for y

Now solve the equation \( x = (y + 2)^2 \) for \( y \). Take the square root of both sides: \( \sqrt{x} = y + 2 \).
04

Isolate y

To isolate \( y \), subtract 2 from both sides: \( y = \sqrt{x} - 2 \). This is the inverse function, but we must ensure that \( x \geq 0 \) to retain the domain of the inverse function.
05

Write the Inverse Function

The inverse function is \( f^{-1}(x) = \sqrt{x} - 2 \) for \( x \geq 0 \).

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!

Key Concepts

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

Domain of a Function
When dealing with functions, understanding the domain is crucial, as it tells us which values are acceptable for the function's input. In the given exercise, our function is defined as \( f(x) = (x + 2)^2 \) with a domain of \([-2, \infty)\). This means \( x \) can be any number from \(-2\) onwards.
The domain is important for two main reasons:
  • It ensures that every value in the domain will produce a valid output with the function.
  • It often affects the calculation of the inverse function, as seen in this problem.
When we found the inverse, the domain of the original function transformed into the range of our inverse function. The inverse domain checks occur when it's essential to maintain consistency with the function's original input restrictions.
Square Root
The square root is key in finding an inverse for functions involving squares. When we have \( x = (y + 2)^2 \), taking the square root of both sides allows us to move closer to isolating \( y \). Understanding square roots means recognizing that:
  • The square root of a number \( x \) is denoted as \( \sqrt{x} \).
  • It represents a value which, when multiplied by itself, gives \( x \).
In this problem, we use this concept to separate parts of the original function's equation. However, we need to be careful 鈥� square roots typically have both positive and negative solutions. Here, since \( x \geq 0 \) in the domain of the inverse, we stick with just the positive root. This results in the expression \( \sqrt{x} \) simplifying our path to finding \( y \).
Isolating Variables
Isolating the variable means expressing one variable explicitly in terms of others. In our exercise, we start with the equation \( x = (y + 2)^2 \). To find the inverse, we want to solve for \( y \). This involves a few steps:
  • First, take the square root of both sides, moving from \( x = (y + 2)^2 \) to \( \sqrt{x} = y + 2 \).
  • Next, subtract 2 from both sides to finally isolate \( y \) as \( y = \sqrt{x} - 2 \).
These manipulations help us rewrite the original function equation to accurately reflect what happens in reverse. We get an expression now defined as a function of \( x \), providing an inverse. Ensuring every step maintains correctness and domain constancy is vital, thus resulting in a usable inverse function \( f^{-1}(x) = \sqrt{x} - 2 \) with its own domain restrictions.

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

Solve the following equations over the complex number system. $$x^{2}+2 x+10=0$$

Use the given length and area of a rectangle to express the width algebraically. Length is \(x+5,\) area is \(2 x^{2}+9 x-5.\)

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation. $$3 x^{3}+11 x^{2}+8 x-4=0$$

Use synthetic division to find the quotient $$\left(x^{4}-12 x^{3}+54 x^{2}-108 x+81\right) \div(x-3)$$

Use synthetic division to find the quotient $$\left(6 x^{3}-x^{2}+5 x+2\right) \div(3 x+1)$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.