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Chapter 2: Problem 26

Identify the set of values \(x\) for which \(y\) will be a real number. $$ y=\frac{2}{x+7} $$

Short Answer

Expert verified
For \(x eq -7\), \(y\) will be a real number.

Step by step solution

01

Identify the Denominator

The denominator of the given function is \(x + 7\).
02

Determine When the Denominator is Zero

The function \(y\) will be undefined when the denominator equals zero. Set \(x + 7\) equal to zero and solve for \(x\).
03

Solve for x

Solve the equation \(x + 7 = 0\). Subtract 7 from both sides to get \(x = -7\).
04

Identify Real Number Condition

Since \(x = -7\) is the value that makes the function undefined, \(y\) will be a real number for all values of \(x\) except \(x = -7\).

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Key Concepts

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

Understanding the Domain of Rational Functions
To solve the problem, we first need to understand what the domain of a rational function is. The domain of a function is the set of all possible input values (in this case, values of \(x\)) that allow the function to produce a real number output. For rational functions, the main restriction comes from the denominator. Any value that makes the denominator zero will cause the function to be undefined. Therefore, we need to ensure that the denominator is never zero for the function to be defined.
Identifying Undefined Values in Algebra
Undefined values in algebra occur when we try to perform an operation that isn鈥檛 allowed. With rational functions, division by zero is not allowed because it leads to an undefined value.
To identify these values, we need to set the denominator equal to zero and solve the equation.
For the function in question, the denominator is \(x + 7\). Setting \(x + 7 = 0\) and solving for \(x\) helps us find the value that makes the function undefined.
In this case, \(x = -7\). This is the value that makes the function undefined.
Exploring Real Number Conditions in Functions
Understanding real number conditions means knowing when the function produces real values as outputs.
For our rational function \(y = \frac{2}{x + 7}\), it will produce real numbers for all values of \(x\) except when \(x\) equals \(-7\).
This is because any other value of \(x\) will result in a non-zero denominator, ensuring the output is a real number.
Thus, the function \(y\) is defined and real for all \(x eq -7\).
Solving for Variables in Algebra
Solving for variables in algebra involves isolating the variable on one side of the equation. We solve for the variable \(x\) to find conditions that affect our function.
In our exercise, we set the denominator \(x + 7\) to zero:
\(x + 7 = 0\).
To isolate \(x\), we subtract 7 from both sides:
\(x = -7\).
This tells us \(x = -7\) is not part of the domain, as it makes the function undefined.
Understanding how to solve such equations is fundamental to working effectively with rational functions.

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Most popular questions from this chapter

Refer to the functions \(f\) and \(g\) and evaluate the functions for the given values of \(x\). \(f=\\{(2,4),(6,-1),(4,-2),(0,3),(-1,6)\\} \quad\) and \(\quad g=\\{(4,3),(0,6),(5,7),(6,0)\\}\) $$(g \circ f)(2)$$

A function is given. (See Examples \(4-5)\) a. Find \(f(x+h)\). b. Find \(\frac{f(x+h)-f(x)}{h}\). $$f(x)=5 x+9$$

Suppose that the average rate of change of a continuous function between any two points to the left of \(x=a\) is negative, and the average rate of change of the function between any two points to the right of \(x=a\) is positive. Does the function have a relative minimum or maximum at \(a\) ?

Use a graphing utility to graph the piecewise-defined function. $$ g(x)=\left\\{\begin{array}{ll} -3.1 x-4 & \text { for } x<-2 \\ -x^{3}+4 x-1 & \text { for } x \geq-2 \end{array}\right. $$

Use a graphing utility to approximate the relative maxima and relative minima of the function on the standard viewing window. Round to 3 decimal places. $$ g(x)=0.4 x^{2}-3 x-2.2 $$
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