/*! 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 14 Find the domain of each function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 14

Find the domain of each function. $$h(x)=\frac{5}{\frac{4}{x}-1}$$

Short Answer

Expert verified
The domain of the function \(h(x)=\frac{5}{\frac{4}{x}-1}\) is all real numbers except 4, or \(x \neq 4\).

Step by step solution

01

Identify the function

The function given is \(h(x) = \frac{5}{\frac{4}{x}-1}\). In this case, you can only divide by non-zero numbers. Therefore, when identifying the domain of this function, it is essential to set up an equation that shows when the denominator equals zero.
02

Set up an equation

Set up an equation equal to zero, so that the function is undefined. So you have to solve the equation setting the denominator equal to zero, like so: \(\frac{4}{x}-1=0\)
03

Solve the equation

Solve the equation \(\frac{4}{x}-1=0\) for 'x'. First, add 1 to both sides to get \(\frac{4}{x}=1\). Now, multiply both sides by 'x' to isolate 'x': \(4=x\). So, 'x' equals 4.
04

Finalize the domain

The function is undefined at 'x' equals 4. So, the domain of the function is all real numbers except 4. Hence, the solution is \(x \neq 4\)

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.

Rational Function
A rational function is a type of function expressed as the ratio of two polynomials. In simpler terms, it means that you have one polynomial divided by another. Polynomials themselves are mathematical expressions involving variables with whole number exponents, and constants. Here are a few things to remember about rational functions:
  • The numerator and the denominator are both polynomials.
  • The most common form is \( R(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
  • An important aspect is ensuring that the denominator \( Q(x) \) cannot be zero, as this would make the function undefined.
In our given example with \( h(x) = \frac{5}{\frac{4}{x}-1}\), we can see that it's a bit more complex due to the fraction inside the denominator. The task is to determine the values that would cause the denominator to become zero, hence making the function undefined.
Undefined Expression
An expression is considered undefined for values of the variable that cause division by zero. In math, division by zero is considered impossible, as it doesn't produce a real number result. When we find that an expression is undefined, it helps us identify the restrictions on the domain of a function.
  • The expression became undefined when the denominator is zero.
  • In the exercise, \( \frac{4}{x}-1 \) is the denominator, and we set it equal to zero to find the undefined condition.
  • Solving \( \frac{4}{x}-1=0 \), we find that \( x=4 \) results in an undefined denominator.
Thus, any value of \( x \) that makes the denominator zero is excluded from the domain of the function, ensuring the function is defined for all other values.
Precalculus
Precalculus involves the study of mathematical concepts that prepare students for calculus. It often includes topics like algebra, trigonometry, and functions. Understanding the domain of a function is a crucial part of precalculus, as it outlines the set of input values (\( x \) values) for which the function is defined.In the context of rational functions, finding the domain involves:
  • Identifying any constraints, like denominators that could become zero.
  • Setting the problematic part of the function (usually the denominator) equal to zero to solve for the excluded values.
  • Expressing the domain as all real numbers except the value(s) that cause the denominator to be zero.
The skills acquired in precalculus, like dealing with undefined expressions, form the foundation needed to tackle more advanced calculus concepts where limits and continuity play a role.

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 by the quadratic formula: \(5 x^{2}-6 x-8=0\).

Determine whether each relation is a function. Give the domain and range for each relation. a. \(\\{(1,6),(1,7),(1,8)\\}\) b. \(\\{(6,1),(7,1),(8,1)\\}\) (Section \(1.2,\) Example 2 )

Will help you prepare for the material covered in the next section. Consider the function defined by $$\\{(-2,4),(-1,1),(1,1),(2,4)\\}$$ Reverse the components of each ordered pair and write the resulting relation. Is this relation a function?

The function \(C(t)=20+0.40(t-60)\) describes the monthly cost, \(C(t),\) in dollars, for a cellphone plan for \(t\) calling minutes, where \(t>60 .\) Find and interpret \(C(100)\).

In Exercises \(53-64,\) complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}-6 y-7=0$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.