/*! 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 93 Find the domain of the rational ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 93

Find the domain of the rational function \(f(x)=\frac{2 x-3}{x-5}\)

Short Answer

Expert verified
The domain is \((-\infty, 5) \cup (5, \infty)\).

Step by step solution

01

- Identify the Denominator

To find the domain of the rational function, start by identifying the denominator. The function given is \(f(x) = \frac{2x-3}{x-5}\). The denominator here is \(x-5\).
02

- Set the Denominator Not Equal to Zero

Rational functions are undefined where their denominators are equal to zero. Set the denominator equal to zero and solve for \(x\): \(x - 5 = 0\).
03

- Solve the Equation

Solve the equation \(x - 5 = 0\) to find the value of \(x\) that makes the denominator zero. \(x = 5\).
04

- State the Domain

The domain of the function includes all real numbers except the value that makes the denominator zero. Since \(x = 5\) makes the denominator zero, the domain is all real numbers except \(x = 5\). In interval notation, this is written as \((-\infty, 5) \cup (5, \infty)\).

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.

headline of the respective core concept
Rational functions are fractions where both the numerator and the denominator are polynomials. In simpler terms, it's an equation that looks like a fraction but with more complex expressions. For example, in the rational function given:
\[ f(x) = \frac{2x-3}{x-5} \]
The numerator is \[2x-3\] and the denominator is \[x-5\].
Understanding rational functions is important because they can have certain values that make them undefined, which directly affects their domain.
headline of the respective core concept
The denominator in a rational function is the value under the division line, which determines when the function is undefined. A function is undefined when the denominator equals zero because division by zero is not possible.
Identifying the denominator is crucial for finding the domain. In our function \[ f(x) = \frac{2x-3}{x-5} \] the denominator is \[x-5\].
The next step is to set the denominator not equal to zero and solve for the variable. This means:
  • Set the denominator equal to zero: \[ x-5 = 0 \]
  • Solve for \[x\]: \[ x = 5 \]
So, \[x=5\] is the value that makes the denominator zero, making the function undefined.
headline of the respective core concept
Interval notation is a way to express the domain of a function in a concise form. It shows which values a function can take (include) and which it cannot (exclude).
For the rational function \[ f(x) = \frac{2x-3}{x-5} \], we found that \(x=5\) makes the denominator zero and thus must be excluded from the domain.
In interval notation, we express the domain by writing the ranges of values excluding \(x=5 \).
  • To denote all real numbers less than 5, we write \[(-\infty, 5)\]
  • To denote all real numbers greater than 5, we write \[(5, \infty)\]
Combining them, the domain of \( f(x)\) in interval notation is \[(-\infty, 5) \cup (5, \infty)\]. This means all real numbers except 5 are in the function's domain.

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

Find the rectangular coordinates of the point with the polar coordinates \(\left(12,-\frac{\pi}{3}\right)\)

Prove that the hyperbola \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\) has the two oblique asymptotes \(y=\frac{a}{b} x\) and \(y=-\frac{a}{b} x\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the area of the triangle: \(a=14, b=11, C=30^{\circ}\)

Use a graphing utility to graph the plane curve defined by the given parametric equations. \(x(t)=4 \sin t+2 \sin (2 t)\) \(y(t)=4 \cos t+2 \cos (2 t)\)

Solve: \((x+3)^{2}=20\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure 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.