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Chapter 3: Problem 81

Find the domain of \(f\). $$f(x)=\frac{5}{x-3}$$

Short Answer

Expert verified
The domain of the function is \( x eq 3 \) or \( (-\infty, 3) \cup (3, \infty) \).

Step by step solution

01

Identify the Function Type

The given function is a rational function, which has the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.
02

Determine the Denominator

For the function \( f(x)=\frac{5}{x-3} \), identify the denominator. Here, the denominator is \( x-3 \).
03

Find Undefined Values

Rational functions are undefined where the denominator is zero. Set the denominator equal to zero and solve for \( x \): \( x - 3 = 0 \).
04

Solve for \( x \)

Solving the equation \( x - 3 = 0 \) gives \( x = 3 \). This is the value that makes the function undefined.
05

State the Domain

The domain of the function \( f(x)=\frac{5}{x-3} \) is all real numbers except where the function is undefined. Hence, the domain is \( x eq 3 \). In interval notation, this is \( (-\infty, 3) \cup (3, \infty) \).

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Key Concepts

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

Understanding Rational Functions
A rational function is a fraction of two polynomials. For example, our given function is a rational function: \[ f(x)=\frac{5}{x-3} \]Here, the numerator is 5, a constant polynomial, and the denominator is \ x - 3 \, another polynomial.Rational functions can behave quite differently than simple polynomials. They can have vertical asymptotes, which are lines that the graph approaches but never touches, and holes, at the points where the function is undefined.Understanding how to handle these will help you graph the function and analyze its behavior.
What Are Polynomials?
Polynomials are expressions that consist of variables and coefficients connected by addition, subtraction, and multiplication. Here are a few examples:
  • 3x + 4
  • x^2 - 2x + 1
  • 5
In general, a polynomial can look something like this: \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \ ... + a_1x + a_0 \]Where \ a_n, a_{n-1}, ...,a_0 are constants.Rational functions use polynomials for both their numerators and denominators. The function \ f(x) = \frac {5}{x-3} \ uses a simple polynomial (just 5) in the numerator and \ x-3 \ which is also a simple polynomial, in the denominator. Understanding polynomials helps make sense of rational functions which often appear in calculus and advanced math.
Handling Undefined Values
Rational functions can be undefined where the denominator equals zero. For the function \[ f(x) = \frac{5}{x-3} \], we set the denominator \ x-3 \ equal to zero to find any undefined points.Solving \ x-3=0 \, we find that when \ x=3 \, the function is undefined.This is crucial because it tells us where the function cannot exist. For our function: \ f(x) = \frac{5}{x-3} \ , it cannot be defined at \ x = 3 \. This leads us to the conclusion about its domain: it includes all real numbers except for 3.To write this in interval notation, we exclude 3:
  • Domain: \ (-\infty, 3) \cup (3, \infty)\
.Knowing how to find these undefined values will help you solve and graph any rational function.

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