Problem 10 - Elementary and Intermediate Algebra

Question

If \(f(x)=x^{2}+3\) and \(g(x)=x-4,\) then \((f / g)(x)=\frac{x^{2}+3}{x-4}\) a. What value of \(x\) makes \(g(x)=0 ?\) b. Fill in the blank: The domain of \(f / g\) is \((-\infty, 4)\) ____ \((4, \infty)\)

Step-by-Step Solution

Answer

a. x = 4; b. The domain is \((- fty, 4) \cup (4, \infty)\).

Step 1: Identify when g(x)=0

To find the value of \(x\) that makes \(g(x)=0\), we solve the equation \(g(x) = x - 4 = 0\).

Step 2: Solve for x

Solving \(x - 4 = 0\) gives us \(x = 4\). Hence, \(g(x) = 0\) when \(x = 4\).

Step 3: State the domain issue

The expression for \(\left(\frac{f}{g}\right)(x) = \frac{x^2 + 3}{x - 4}\) is undefined when the denominator is zero, i.e., when \(x = 4\).

Step 4: Determine the domain

Since the function \(\frac{f}{g}\) is undefined at \(x = 4\), the domain excludes \(4\). Therefore, the domain is \((-\infty, 4) \cup (4, \infty)\).

Key Concepts

Function Operations

When we work with functions, performing operations such as addition, subtraction, multiplication, and division are a common task. These operations are essential in creating new functions from existing ones. For instance, in the given exercise, we are dealing with the division of two functions:

• The function \(f(x) = x^2 + 3\).
• The function \(g(x) = x - 4\).

To perform the division operation, represented as \( (f / g)(x) = \frac{f(x)}{g(x)} \), we substitute the definitions of \(f\) and \(g\) into the expression, resulting in \( (f / g)(x) = \frac{x^2 + 3}{x - 4}\).
This function gives us a rational function. Function operations help expand our understanding of how different functions interact when combined, forming the basis for more complex mathematical modelling.

Domain Restrictions

Domains define the set of possible input values (usually \(x\)) that a function can accept to produce a valid output. However, certain functions come with restrictions. A common restriction occurs with rational functions because it's important that the denominator never equals zero.
Let's take the function \( (f / g)(x) = \frac{x^2 + 3}{x - 4}\), derived from our exercise:

• For this function, we cannot divide by zero. Therefore, \(x - 4\) should not be zero.
• Solving \(x - 4 = 0\), we find \(x = 4\) causes an issue.

This means at \(x = 4\), the function "breaks," so we exclude this from the domain.
The domain of \(f / g\) is written as \((-7infty, 4) 2cup (4, 7infty)\), meaning all real numbers except \(x = 4\).
Understanding domain restrictions is key in ensuring our functional operations are valid and meaningful.

Rational Function

A rational function is a type of function expressed as the ratio of two polynomials. It takes the form \( \frac{P(x)}{Q(x)} \) where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not zero. For example, in our exercise, we see:

• The numerator polynomial \(P(x) = x^2 + 3\).
• The denominator polynomial \(Q(x) = x - 4\).

Rational functions often have domain restrictions because setting the denominator equal to zero would result in an undefined expression.
In this context, this results in a typical challenge of determining the points where the function might "disappear."
Rational functions can display interesting behavior such as vertical asymptotes, which occur at the points of their domain restrictions, like \(x = 4\) in this example.
Exploring rational functions offers deep insights into polynomial interactions and their graphical characteristics.