Problem 15 - College Algebra

Question

For the following exercises, find the domain, vertical asymptotes, and horizontes of the functions. $$ f(x)=\frac{3 x-4}{x^{3}-16 x} $$

Step-by-Step Solution

Answer

Domain: all real numbers except 0, 4, -4; Vertical asymptotes: x = 0, 4, -4; Horizontal asymptote: y = 0.

Step 1: Identify Domain

To find the domain of the function \( f(x) = \frac{3x-4}{x^3 - 16x} \), we need to identify where the denominator is zero since division by zero is undefined. Set the denominator equal to zero: \( x^3 - 16x = 0 \). Factor the denominator as \( x(x^2 - 16) = 0 \), which simplifies to \( x(x-4)(x+4) = 0 \). The roots are \( x = 0, 4, -4 \). Therefore, the domain of \( f(x) \) is all real numbers except \( x = 0, 4, -4 \).

Step 2: Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero but the numerator is not zero. From Step 1, the denominator zeros are \( x = 0, 4, -4 \). Check these in the numerator \( 3x-4 \): \( 3(0)-4 = -4 \), \( 3(4)-4 = 8 \), and \( 3(-4)-4 = -16 \). None of these zeros make the numerator equal to zero, so vertical asymptotes are at \( x = 0, 4, -4 \).

Step 3: Find Horizontal Asymptotes

For horizontal asymptotes of \( f(x) = \frac{3x-4}{x^3 - 16x} \), compare the degrees of the numerator and denominator. The degree of the numerator is 1 and the degree of the denominator is 3. When the degree of the numerator is less than the denominator, the horizontal asymptote is \( y = 0 \).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (or "x" values) that the function can accept without any issues such as division by zero or square roots of negative numbers. When working with rational functions, such as \(f(x) = \frac{3x-4}{x^3 - 16x}\), finding the domain involves identifying values of \(x\) that cause the denominator to be zero.
To find these values, set the denominator equal to zero:

• Factor the expression: \(x^3 - 16x = x(x-4)(x+4) = 0\)
• Determine the roots: \(x = 0\), \(x = 4\), and \(x = -4\)

These roots are not allowed in the domain because they make the denominator zero, which causes division by zero—an undefined operation in mathematics. Thus, the domain of \(f(x)\) is all real numbers except \(x = 0\), \(x = 4\), and \(x = -4\). More formally, it's expressed as:
\(\{x \in \mathbb{R} \mid x eq 0, 4, -4\}\)

Vertical Asymptotes

Vertical asymptotes are lines where the function grows without bound as \(x\) approaches a certain value. These typically occur where the denominator is zero, given that the numerator is not zero at these points.
For the function \(f(x) = \frac{3x-4}{x^3 - 16x}\), the denominator zeroes were found earlier: \(x = 0\), \(x = 4\), and \(x = -4\). To confirm these vertical asymptotes, ensure the numerator \(3x-4\) is not zero at these points:

• \(3(0) - 4 = -4 eq 0\)
• \(3(4) - 4 = 8 eq 0\)
• \(3(-4) - 4 = -16 eq 0\)

Since the numerator is not equal to zero at these points, vertical asymptotes indeed exist at \(x = 0\), \(x = 4\), and \(x = -4\). These asymptotes represent points where the function dramatically shoots upwards or downwards as \(x\) approaches the asymptotic value.

Horizontal Asymptotes

Horizontal asymptotes show the behavior of a function as \(x\) approaches infinity or negative infinity. They indicate a line that the graph of the function approaches but never quite reaches.
To determine the horizontal asymptote of a function, compare the degrees of the numerator and the denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
• If they are equal, the horizontal asymptote is \(y = \text{leading coefficient of numerator} \div \text{leading coefficient of denominator}\).
• If the degree of the numerator is greater, there is no horizontal asymptote.

In the case of \(f(x) = \frac{3x-4}{x^3 - 16x}\), the degree of the numerator is 1, and the degree of the denominator is 3. Here, the degree of the numerator is less, so the horizontal asymptote is \(y = 0\). This means as \(x\) moves towards infinity, the graph of the function approaches the line \(y = 0\).