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Chapter 2: Problem 87

Find the domain of the function given by each equation. $$ f(x)=\frac{3}{8-5 x} $$

Short Answer

Expert verified
The domain is all real numbers except \( x = \frac{8}{5} \), or \[ (-\infty, \frac{8}{5}) \cup (\frac{8}{5}, +\infty) \].

Step by step solution

01

Identify the Denominator

The given function is \[ f(x) = \frac{3}{8 - 5x} \] Identify the denominator of the function, which is the part of the fraction that is below the division line. In this case, the denominator is \( 8 - 5x \).
02

Determine the Undefined Points

The function will be undefined when the denominator is zero because division by zero is undefined. Set the denominator equal to zero and solve for \( x \): \[ 8 - 5x = 0 \]
03

Solve for \( x \)

Solve the equation \( 8 - 5x = 0 \) to find the value of \( x \) that makes the denominator zero: \[ 8 - 5x = 0 \]\[ -5x = -8 \]\[ x = \frac{8}{5} \]
04

Define the Domain

Since the denominator is zero at \( x = \frac{8}{5} \), the function is undefined at this point. Therefore, the domain of the function includes all real numbers except \( x = \frac{8}{5} \). The domain in interval notation is: \[ (-\infty, \frac{8}{5}) \cup (\frac{8}{5}, +\infty) \]

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Key Concepts

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

Rational Functions
A rational function is a function of the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \ P(x) \ and \ Q(x) \ are polynomials and \ Q(x) \ is not zero. In simpler terms, it's a fraction where the numerator and the denominator are polynomials.
For example, \( f(x) = \frac{2x + 3}{x^2 - 4} \) is a rational function, with a polynomial of \ 2x + 3 \ in the numerator and \ x^2 - 4 \ in the denominator.
Rational functions often have special characteristics:
  • They can have horizontal and vertical asymptotes.
  • They can be undefined at certain points
  • Their domain is determined by the values of \ x \ that make the denominator non-zero.
Undefined Points
Undefined points in a rational function occur where the denominator, \ Q(x) \ equals zero. Division by zero is impossible in mathematics, so these points do not exist in the function.
To find the undefined points:
  • Set the denominator equal to zero.
  • Solve the equation to find the value(s) of \ x \ that make the denominator zero.
For example, if \ f(x) = \frac{3}{8 - 5x} \, then set \ 8 - 5x = 0 \ to find the value of \ x \. Solving this, we get:
\[ 8 - 5x = 0 \]
\[ -5x = -8 \]
\[ x = \frac{8}{5} \]
Thus, \ x = \frac{8}{5} \ is the undefined point for the function.
Always remember to exclude these values when determining the domain of a rational function.
Interval Notation
Interval notation is a way of specifying a range of values on a number line. It describes the domain or the set of values that a variable can take.
There are a few key symbols in interval notation:
  • ( or ): Use a parenthesis when the value is not included in the interval.
  • [ or ]: Use a bracket when the value is included in the interval.
  • \ (-\infty, +\infty) \: Denotes all real numbers from negative to positive infinity.
  • U: Represents the union of two disjoint intervals.
In our example, \ f(x) = \frac{3}{8-5x} \, is undefined at \ x = \frac{8}{5} \. So, the domain of \ f(x) \ includes all real numbers except \ x = \frac{8}{5} \. Using interval notation, we express it as:
\[ (-\infty, \frac{8}{5}) \cup (\frac{8}{5}, +\infty) \]
This means all real numbers except \ \frac{8}{5} \. Think of it as combining all numbers less than \ \frac{8}{5} \ with all numbers greater than \ \frac{8}{5} \.

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