Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the domain of \(f(x)=\frac{5 x-1}{x^{3}-16 x}\).

Short answer

Domain: \((-\infty, -4) \cup (-4, 0) \cup (0, 4) \cup (4, \infty)\)

Step-by-step solution

- Identify the Denominator

First, focus on the denominator of the function, which is given by the expression in the denominator: \( x^3 - 16x \).

- Factorize the Denominator

Factor the denominator completely by taking out the common factor.\( x^3 - 16x = x(x^2 - 16) = x(x - 4)(x + 4). \)

- Set the Denominator Equal to Zero

Set the factored denominator equal to zero to find the values of x for which the denominator is zero.\( x(x - 4)(x + 4) = 0 \)

- Solve for x

Solve the equation from Step 3 to find the values of x:\( x = 0, \; x - 4 = 0 \Rightarrow x = 4, \; x + 4 = 0 \Rightarrow x = -4 \)

- Determine the Excluded Values

The values of x found in Step 4 make the denominator zero, so they must be excluded from the domain of the function. These values are:\( x = 0, x = 4, \; \text{and} \; x = -4 \)

- State the Domain

The domain of the function is all real numbers except the excluded values. In interval notation, this is written as:\( \text{Domain} = (-\infty, -4) \cup (-4, 0) \cup (0, 4) \cup (4, \infty) \)

Key concepts

rational functions

A rational function is a ratio of two polynomials. It is in the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). Because the denominator \( Q(x) \) cannot be zero, we need to find and exclude these values from the domain of the function.Notice that for rational functions, when the denominator is zero, the function is undefined at those points.
Examples of rational functions include:

• \( f(x) = \frac{1}{x-2} \) which is undefined for \( x = 2 \)
• \( g(x) = \frac{2x+3}{x^2 - 9} \) which is undefined at \( x = 3 \) and \( x = -3 \)

factoring polynomials

Factoring polynomials is crucial in simplifying expressions and solving equations.To factor a polynomial means to write it as a product of simpler polynomials.For example, \( x^3 - 16x \) can be factored by first taking out the common factor \( x \).Thus, we get:
\( x^3 - 16x = x(x^2 - 16) \)
Next, notice that \( x^2 - 16 \) is a difference of squares, which can be written as:
\( x^2 - 16 = (x - 4)(x + 4) \)
Hence, the completely factored form is:
\( x(x - 4)(x + 4) \)
Factoring makes it easier to identify the roots or zeros of the function. These roots are key in determining the domain.

excluded values

When dealing with rational functions, excluded values are the x-values that make the denominator zero.Identifying excluded values is essential because these values make the function undefined.Consider the rational function
\( f(x) = \frac{5x - 1}{x^3 - 16x} \)
First, we factor the denominator as explained:[br] \( x^3 - 16x = x(x - 4)(x + 4) \)
By setting the factored denominator equal to zero:
\( x(x - 4)(x + 4) = 0 \), we solve for x:
\( x = 0, x = 4, x = -4 \)
These values are excluded from the domain because they would make the denominator zero, which is not allowed in rational functions.

interval notation

In representing the domain of a function, interval notation is a shorthand way to express the set of all valid input values.Interval notation uses parentheses \((...)\) for values that are not included and brackets \([...]\) for values that are included.For example, the domain excluding values \( x = 0, x = 4, \text{and} \ x = -4 \) is written as:
\( (-\infty, -4) \cup (-4, 0) \cup (0, 4) \cup (4, \infty) \)
This tells us that all numbers are part of the domain, except \(-4, 0, \text{and } 4\). Using interval notation helps in easily representing complex domains without listing every single number.