Textbook Question
Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies directly as x. y = 65 when x = 5. Find y when x = 12.
172
views
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 1
Find the domain of each rational function. f(x)=5x/(x−4)
Verified step by step guidance1
Identify the rational function given: \(f(x) = \frac{5x}{x - 4}\).
Recall that the domain of a rational function includes all real numbers except where the denominator is zero, because division by zero is undefined.
Set the denominator equal to zero to find the values to exclude: \(x - 4 = 0\).
Solve the equation for \(x\): \(x = 4\).
Conclude that the domain of \(f(x)\) is all real numbers except \(x = 4\), which can be written in interval notation as \((-\infty, 4) \cup (4, \infty)\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Functions
A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding rational functions involves recognizing that the function is undefined where the denominator Q(x) equals zero, which affects the domain.
Recommended video:
6:04
Intro to Rational Functions
Domain of a Function
The domain of a function is the set of all input values (x-values) for which the function is defined. For rational functions, the domain excludes values that make the denominator zero, since division by zero is undefined.
Recommended video:
3:51Domain Restrictions of Composed Functions
Finding Values that Make the Denominator Zero
To find the domain of a rational function, set the denominator equal to zero and solve for x. These values are excluded from the domain because they cause division by zero, which is undefined in mathematics.
Recommended video:
03:42Finding Zeros & Their Multiplicity