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Chapter 2: Problem 6

Find the domain of each rational function. $$f(x)=-\frac{2 x}{x^{2}+9}$$

Short Answer

Expert verified
The domain of the function is all real numbers, \( x \in (-\infty, \infty) \).

Step by step solution

01

Identify the Denominator

The denominator of the rational function is the expression that determines the values of \( x \) that make the function undefined. For the given function \( f(x) = -\frac{2x}{x^2 + 9} \), the denominator is \( x^2 + 9 \).
02

Solve for Domain Constraints

A rational function is undefined when its denominator is zero. So, set the denominator equal to zero and solve for \( x \): \( x^2 + 9 = 0 \).
03

Solve the Denominator Equation

Solving \( x^2 + 9 = 0 \), we get \( x^2 = -9 \). However, the square of any real number cannot be negative, meaning there is no real solution for \( x^2 = -9 \).
04

Identify the Domain

Since the equation \( x^2 = -9 \) has no real solution, the denominator \( x^2 + 9 \) is never zero for any real \( x \). Thus, the domain of the function is all real numbers, \( x \in (-\infty, \infty) \).

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Key Concepts

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

Rational Functions
Rational functions are a type of function expressed as the ratio of two polynomials. Imagine a fraction where both the numerator and the denominator are polynomials.
A simple example is \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomial expressions.

The behavior of a rational function largely depends on its denominator because that's where things get interesting.
  • If the denominator equals zero, the function becomes undefined because division by zero is impossible mathematically.
  • To explore the domain of a rational function, we focus on ensuring the denominator never hits zero for real values of \( x \).
This understanding is central to exploring and determining the domain of any rational function.
Real Numbers
Real numbers encompass most numbers you encounter in everyday life. They include whole numbers, fractions, and decimals, as well as positives, negatives, and zero.
In the context of solving expressions or understanding domains, real numbers are crucial.
  • Unlike imaginary numbers, which include imaginary units like \( i \) (where \( i^2 = -1 \)), real numbers don't show up in equations where a square equals a negative number.
  • This means when solving an equation, if we land on a solution like \( x^2 = -9 \), within the realm of real numbers, we conclude that no solution exists.
So, when determining the domain of a function, real numbers guide us in understanding which values \( x \) can realistically take.
Undefined Expressions
In mathematics, an expression becomes undefined when an operation cannot be performed within accepted mathematical rules.
A common scenario for undefined expressions is division by zero.
  • In rational functions, these undefined points occur when the denominator is zero, leading to an undefined function output.
  • To find these points, we set the denominator equal to zero and solve for \( x \).

In our exercise, we faced the equation \( x^2 + 9 = 0 \). Solving, it led us to \( x^2 = -9 \).
Since no real number squared gives a negative number, "undefined" here helps us realize that the rational function \(-\frac{2x}{x^2+9}\) is actually valid across all real numbers, precisely because those conditions are never met.
This provides the key insight that, for this function, the domain covers all real numbers.

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Most popular questions from this chapter

Write a rational function that has vertical asymptotes at \(x=-3\) and \(x=1\) and oblique asymptote \(y=3 x, y\) -intercept \((0,2), \text { and } x \text { -intercept ( } 2,0) .\) Round your answers to two ferimal nlace

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