/*! 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 27 Evaluate (if possible) the funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 27

Evaluate (if possible) the function at each specified value of the independent variable and simplify. \(q(x)=1 /\left(x^{2}-9\right)\) (a) \(q(0)\) (b) \(q(3)\) (c) \(q(y+3)\)

Short Answer

Expert verified
The solutions are: (a) \(q(0) = -1/9\), (b) \(q(3)\) is undefined, and (c) \(q(y+3) = 1 /(y^2 + 6y)\).

Step by step solution

01

Substitute x = 0 in the function

To find \(q(0)\), we substitute \(x=0\) into the equation \(q(x)=1 /\left(x^{2}-9\right)\). This gives us \(q(0)= 1 /(0^2 - 9) = -1/9.\
02

Substitute x = 3 in the function

To find \(q(3)\), we substitute \(x=3\) into the equation \(q(x)=1 /\left(x^{2}-9\right)\). This results in a denominator equal to zero (since \(3^2 - 9 = 0\)), so \(q(3)\) is undefined.
03

Substitute \(x = y + 3\) in the function

To compute \(q(y+3)\), we substitute \(x = y + 3\) into the equation, which gives us \(q(y+3) = 1 /((y+3)^2 - 9). To simplify this, it can be written as 1 /((y^2 + 6y + 9) - 9) = 1 /(y^2 + 6y).

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.

Rational Expressions
In mathematics, rational expressions are fractions that involve polynomials in their numerator, denominator, or both. Similar to how we work with numerical fractions, with rational expressions, we aim to simplify them as much as possible.

Let's examine the function given in the exercise, q(x) = 1/(x^2 - 9). Here, the expression consists of a polynomial in the denominator, making q(x) a rational expression. Simplifying such expressions involves factoring polynomials and cancelling common factors, but there's a crucial aspect we must not overlook鈥攊dentifying when the expression is undefined, which brings us to our next important concept.
Undefined Values
Rational expressions can encounter undefined values where the denominator equals zero, since division by zero is not allowed in mathematics. It鈥檚 important for students to remember that rational expressions like 1/(x^2 - 9) are undefined for certain values of x that make the denominator zero.

In the example q(3), we see how substituting 3 for x leads to a denominator of zero, rendering the expression undefined. It is critical to always check the denominator before declaring a rational expression as simplified or evaluated.
Substitution Method
The substitution method is a fundamental technique used to evaluate functions, where a specific value is plugged into the place of the variable within an expression. In the exercise, the substitution method is applied three different times with various values of x.

Each substitution step requires careful execution to ensure arithmetic errors are avoided. As shown in step 3, when substituting an expression like (y+3) instead of a single number, it鈥檚 necessary to deal with expanding and simplifying, which could involve factoring or distribution laws.

Understanding the substitution method not only allows students to evaluate functions at given points but also lays the groundwork for more advanced mathematical problem-solving in calculus and beyond.

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

(a) Given that \(y\) varies directly as the square of \(x\) and \(x\) is doubled, how will \(y\) change? Explain. (b) Given that \(y\) varies inversely as the square of \(x\) and \(x\) is doubled, how will \(y\) change? Explain.

The research and development department of an automobile manufacturer has determined that when a driver is required to stop quickly to avoid an accident, the distance (in feet) the car travels during the driver's reaction time is given by \(R(x)=\frac{3}{4} x,\) where \(x\) is the speed of the car in miles per hour. The distance (in feet) traveled while the driver is braking is given by \(B(x)=\frac{1}{15} x^{2}.\) (a) Find the function that represents the total stopping distance \(T.\) (b) Graph the functions \(R, B,\) and \(T\) on the same set of coordinate axes for \(0 \leq x \leq 60.\) (c) Which function contributes most to the magnitude of the sum at higher speeds? Explain.

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=\sqrt{2 x+3}$$

Assume that \(y\) is directly proportional to \(x .\) Use the given \(x\) -value and \(y\) -value to find a linear model that relates \(y\) and \(x .\) $$x=4, y=8 \pi$$

Use the given values of \(k\) and \(n\) to complete the table for the direct variation model \(y=k x^{n} .\) Plot the points in a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|}\hline x & 2 & 4 & 6 & 8 & 10 \\\\\hline y=k x^{n} & & & & & \\\\\hline\end{array}$$ $$k=\frac{1}{4}, n=3$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.