/*! 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 34 Find the specified function valu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 34

Find the specified function value, if it exists. $$ g(x)=-\sqrt{(x+1)^{2}} ; g(-3), g(4), g(-5) $$

Short Answer

Expert verified
g(-3) = -2, g(4) = -5, g(-5) = -4.

Step by step solution

01

Understand the Function

The given function is \( g(x) = -\sqrt{(x+1)^{2}} \). Note that the square of any real number is always non-negative.
02

Simplify the Function

Since the square root and the square cancel each other out, \( \sqrt{(x+1)^{2}} = |x+1| \), where \(|x+1|\) denotes the absolute value of \(x+1\). Thus, the function simplifies to \( g(x) = -|x+1| \).
03

Evaluate \( g(-3) \)

Substitute \(-3\) for \(x\) in the simplified function: \[ g(-3) = -|(-3)+1| = -|-2| = -2 \]
04

Evaluate \( g(4) \)

Substitute \(4\) for \(x\) in the simplified function: \[ g(4) = -|(4)+1| = -|5| = -5 \]
05

Evaluate \( g(-5) \)

Substitute \(-5\) for \(x\) in the simplified function: \[ g(-5) = -|(-5)+1| = -|-4| = -4 \]

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.

absolute value
The absolute value of a number is its distance from zero on the number line. It鈥檚 always non-negative. For example, the absolute value of both -5 and 5 is 5. Mathematically, the absolute value of a number x is denoted as \(|x|\). Here are a few key properties:
  • \(|x| = x\) if \(x \geq 0\)
  • \(|x| = -x\) if \(x < 0\)
In the context of our original function, \( g(x) = -\sqrt{(x+1)^2} \), simplifying it further using the absolute value property gives us \( g(x) = -|x+1| \). This step is crucial for evaluating the function at specific points, as it makes it clear how to handle positive and negative inputs.
function evaluation
Function evaluation involves substituting specific values into a given function to find the output. For instance, with \( g(-3) \) in our function \(-|x+1|\):
  • Substitute \(x\) with \(-3\): \( g(-3) = -|(-3) + 1| \)
  • Simplify inside the absolute value: \( -|(-2)| \)
  • Find the absolute value: \(-2) = 2 \)
  • Apply the negative sign: \(-2\)
Thus, \g(-3) = -2\. This same process applies for other values like \ g(4) \ and \ g(-5) \.
radical expressions
Radical expressions involve roots, like square roots. In \(\text{g(x) = -}\sqrt{(x+1)^2}\), the radical (or square root) and the square cancel each other out. This simplifies to \( -|x + 1| \). It's important to remember:
  • A square root combined with a square, as in \(\sqrt{(x+1)^{2}}\), simplifies to the absolute value: \( |x+1| \).
  • This simplification happens because the square ensures the quantity inside is non-negative.
Remember, the square root function yields only non-negative results, which aligns perfectly with the concept of absolute value.
simplifying expressions
Simplifying expressions makes evaluating functions easier. Our function, \( g(x) = -|x+1| \), is a product of simplifying \(-\sqrt{(x+1)^2}\). Here are key steps:
  • Recognize that the square and square root cancel: \(\sqrt{(x+1)^2} = |x+1|\).
  • Add the negative sign from the function: \( g(x) = -|x+1| \).
This simplification helps in straightforward evaluations, such as \g(-3) = -|-2| = -2\. Once you learn to simplify expressions, evaluating complex functions becomes much more manageable.

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

Simplify. $$\left(\frac{1}{2}-\frac{1}{3} i\right)^{2}-\left(\frac{1}{2}+\frac{1}{3} i\right)^{2}$$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[3]{x^{2} y}(\sqrt{x y}-\sqrt[5]{x y^{3}}) $$

Simplify. $$\frac{i-i^{38}}{1+i}$$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \frac{\sqrt[5]{a^{4} b}}{\sqrt[3]{a b}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \frac{\sqrt[5]{(y-9)^{3}}}{\sqrt{y-9}} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

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.