/*! 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 79 Determine the domain of each rat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 79

Determine the domain of each rational function. $$f(p)=\frac{1}{p-7}$$

Short Answer

Expert verified
The domain of the rational function \(f(p) = \frac{1}{p-7}\) is \(\{ p \in \mathbb{R} \mid p \neq 7 \}\).

Step by step solution

01

Identify the denominator of the rational function

In the given function: $$f(p) = \frac{1}{p-7}$$ The denominator is (p-7).
02

Determine the value(s) of p that make the denominator zero

Set the denominator equal to zero and solve for p: $$p-7 = 0$$ Add 7 to both sides: $$p = 7$$
03

Exclude the value(s) found in Step 2 from the domain

As the function becomes undefined when p=7, we need to exclude this value from the domain of the function. Thus, the domain of f(p) is: $$\{ p \in \mathbb{R} \mid p \neq 7 \}$$

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.

Denominator Analysis
When analyzing rational functions, identifying the denominator is the first key step. The denominator is crucial because it determines where the function might be undefined. In our example, the function is given by \( f(p) = \frac{1}{p-7} \). Here, the denominator is \( p-7 \). Always pay close attention to the denominator, as it is where potential problems, like division by zero, can arise. By inspecting this part of the function closely, we can prevent any possible errors or misunderstandings in the overall function analysis.
Undefined Values
When a denominator equals zero, the function is undefined at these points. For the function \( f(p) = \frac{1}{p-7} \), setting the denominator equal to zero,\[ p - 7 = 0 \]helps us find these undefined points. Solving \( p - 7 = 0 \) gives the value \( p = 7 \), indicating the function is undefined when \( p \) equals 7. Undefined values are critical because they tell us where the function breaks down. By identifying these values, we ensure that the function behaves predictably within its domain.
Solving Equations
To determine the undefined values, we solve the equation obtained from the denominator of the rational function. In our case, the equation \( p-7 = 0 \) is straightforward.- Add 7 to both sides to isolate \( p \): \[ p = 7 \]Through solving these simple equations, we gain insight into where our function cannot operate as intended. Solutions like these serve to exclude certain values and help us define a range for the variable that will keep the function valid.
Function Domain
The function domain represents all the possible input values that keep the function well-defined. After solving the equation \( p-7 = 0 \), we've identified \( p = 7 \) as a point that makes the denominator zero. This point must be excluded from the domain. So, for the function \( f(p) = \frac{1}{p-7} \), the domain includes all real numbers except 7.- We can express the domain as: \( \{ p \in \mathbb{R} \mid p eq 7 \} \)It's important when determining function domains to be thorough, ensuring any values that could cause division by zero are excluded.

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 each equation. $$\frac{w}{5}=\frac{w-3}{w+1}+\frac{12}{5 w+5}$$

Identify the least common denominator of each group of rational expression, and rewrite each as an equivalent rational expression with the LCD as its denominator. \(\frac{10}{4 k-1}, \frac{k}{1-16 k^{2}}\)

Perform the indicated operations. $$\frac{3 b-1}{b^{2}+8 b}+\frac{b}{3 b^{2}+25 b+8}+\frac{2}{3 b^{2}+b}$$

Solve for the indicated variable. $$\frac{1}{t}=\frac{1}{r}-\frac{1}{s} \text { for } r$$

Solve each equation. $$\frac{r-2}{2}=\frac{3 r-5}{4}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.