/*! 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 2 \(f(x)=\sqrt{x^{2}+9}, \quad a=-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 2

\(f(x)=\sqrt{x^{2}+9}, \quad a=-4\)

Short Answer

Expert verified
The value of the function at \(x=-4\) is 5.

Step by step solution

01

Substitution of the Given Value

Substitute the given value \(x=-4\) into the function \(f(x)=\sqrt{x^{2}+9}\). This gives us: \(f(-4)=\sqrt{(-4)^{2}+9}\).
02

Simplification

Simplify the expression under the square root. This gives us: \(f(-4)=\sqrt{16+9} = \sqrt{25}\).
03

Calculating the Square Root

Calculate the square root of 25. This gives us: \(f(-4)=5\) which is the value of the function at \(x=-4\).

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.

Square Roots
Square roots are a fundamental mathematical operation. When you take the square root of a number, you're looking for a value that, when multiplied by itself, gives you the original number. For instance, the square root of 25 is 5 because 5 times 5 equals 25. Square roots are denoted by the symbol \( \, \sqrt{\,} \).

Understanding square roots can help simplify complex algebraic expressions and solve equations. In our exercise, we calculated the square root of 25 to get the final answer. Remember, only non-negative numbers have real-number square roots.
Substitution Method
The substitution method involves replacing a variable in an expression with a given value. This simplifies evaluating functions and understanding relationships between numbers.

In our exercise, substitution was the first step. The variable \( x \) was replaced by the given value \( -4 \). This substituted value transforms the function from a general form, \( f(x) = \sqrt{x^{2}+9} \), into a specific number calculation: \( f(-4) = \sqrt{(-4)^2+9} \).

Using substitution helps make abstract algebraic expressions concrete and easier to solve.
Simplification
Simplification reduces a mathematical expression to its simplest form, making it easier to work with or evaluate. During simplification, you can combine like terms, factor expressions, and perform arithmetic operations to consolidate the expression.

In the context of our exercise, simplification involved reducing the expression under the square root. After substituting \( x = -4 \), the expression \( (-4)^2 + 9 \) was simplified to \( 16 + 9 \) and further to \( 25 \). This step made it possible to calculate the square root accurately and easily.
Algebraic Expressions
Algebraic expressions include variables, constants, and operations (such as addition, subtraction, multiplication, and division). They form the backbone of algebra.

In our example, \( \sqrt{x^2+9} \) is an algebraic expression involving a square root and a variable \( x \). Manipulating such expressions often involves substitution, simplification, and other algebraic methods to evaluate or solve them.

Understanding how to construct and deconstruct algebraic expressions allows you to solve equations efficiently and apply problem-solving skills in various mathematical contexts.

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

sign of \(f^{\prime}\) Assume that \(f\) is differentiable on \(a \leq x \leq b\) and that \(f(\)b\()<$$f$$(\)a\()\). Show that \(f^{\prime}\) is negative at some point between \(a\) and \(b\).

Multiple Choice What is the maximum area of a right triangle with hypotenuse 10? \(\begin{array}{llll}{\text { (A) } 24} & {\text { (B) } 25} & {\text { (C) } 25 \sqrt{2}} & {\text { (D) } 48} & {\text { (E) } 50}\end{array}\)

How We Cough When we cough, the trachea (windpipe) contracts to increase the velocity of the air going out. This raises the question of how much it should contract to maximize the velocity and whether it really contracts that much when we cough. Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average flow velocity \(v(\) in \(\mathrm{cm} / \mathrm{sec})\) can be modeled by the equation $$v=c\left(r_{0}-r\right) r^{2}, \quad \frac{r_{0}}{2} \leq r \leq r_{0}$$ where \(r_{0}\) is the rest radius of the trachea in \(\mathrm{cm}\) and \(c\) is a positive constant whose value depends in part on the length of the trachea. (a) Show that \(v\) is greatest when \(r=(2 / 3) r_{0},\) that is, when the trachea is about 33\(\%\) contracted. The remarkable fact is that \(X\) -ray photographs confirm that the trachea contracts about this much during a cough. (b) Take \(r_{0}\) to be 0.5 and \(c\) to be \(1,\) and graph \(v\) over the interval \(0 \leq r \leq 0.5 .\) Compare what you see to the claim that \(v\) is a maximum when \(r=(2 / 3) r_{0}\) .

Parallel Tangents Assume that \(f\) and \(g\) are differentiable on \([a, b]\) and that \(f(a)=g(a)\) and \(f(b)=g(b) .\) Show that there is at least one point between \(a\) and \(b\) where the tangents to the graphs of \(f\) and \(g\) are parallel or the same line. Illustrate with a sketch.

Production Level Suppose \(c(x)=x^{3}-20 x^{2}+20,000 x\) is the cost of manufacturing \(x\) items. Find a production level that will minimize the average cost of making \(x\) items.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

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.