Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, determine the domain for each function in interval notation. Given \(f(x)=\frac{1}{x-4}\) and \(g(x)=\frac{1}{6-x},\) find \(f+g, \quad f-g, \quad f g,\) and \(\frac{f}{g}\)

Step-by-Step Solution

Answer

The domain for each combination is \((-\infty, 4) \cup (4, 6) \cup (6, \infty)\).

Step 1: Find the domain of f(x)

The function \( f(x) = \frac{1}{x-4} \) is undefined when the denominator is zero. Set \( x - 4 = 0 \) to find when this happens. Solving this gives \( x = 4 \). Thus, the domain of \( f(x) \) is all real numbers except 4, which is expressed in interval notation as \( (-\infty, 4) \cup (4, \infty) \).

Step 2: Find the domain of g(x)

The function \( g(x) = \frac{1}{6-x} \) is undefined when the denominator is zero. Set \( 6 - x = 0 \) to find when this happens. Solving this equation gives \( x = 6 \). So, the domain of \( g(x) \) is all real numbers except 6, which is expressed in interval notation as \( (-\infty, 6) \cup (6, \infty) \).

Step 3: Find the domain of f+g and f-g

The domain of \( f + g \) and \( f - g \) is the intersection of the domains of \( f(x) \) and \( g(x) \). Both functions are undefined at \( x = 4 \) and \( x = 6 \). Therefore, the domain for both \( f+g \) and \( f-g \) is \( (-\infty, 4) \cup (4, 6) \cup (6, \infty) \).

Step 4: Find the domain of f*g

The domain of the product \( f \cdot g = \frac{1}{x-4} \cdot \frac{1}{6-x} \) is also the intersection of their domains, which excludes \( x = 4 \) and \( x = 6 \). So, the domain is \( (-\infty, 4) \cup (4, 6) \cup (6, \infty) \).

Step 5: Find the domain of \( \frac{f}{g} \)

The function \( \frac{f}{g} = \frac{\frac{1}{x-4}}{\frac{1}{6-x}} \) can be rewritten as \( \frac{6-x}{x-4} \). It's undefined where \( g(x) \) is zero (\( x = 6 \)) and also where \( f(x) \) is zero (\( x = 4 \)). Therefore, the domain is \( (-\infty, 4) \cup (4, 6) \cup (6, \infty) \).

Key Concepts

Interval Notation

Interval notation is a way to describe the set of all possible values (or the domain) that a function can accept. In the case of rational functions, which are functions defined by fractions

• like \( f(x) = \frac{1}{x-4} \)
• or \( g(x) = \frac{1}{6-x} \)

we need to exclude specific numbers from their domain. Interval notation allows us to succinctly express these exclusions. For example, if a function is defined for all real numbers except for 4, its domain can be expressed as \( (-\infty, 4) \cup (4, \infty) \). The parentheses indicate that the endpoints are not included in the interval.
This method makes reading and writing possible values for functions concise and clear, especially when multiple exclusions are involved.

Rational Functions

A rational function is any function that can be written in the form of a ratio of two polynomials. For example, the function \( f(x) = \frac{1}{x-4} \) is a simple rational function.
In these types of functions, the denominator must not be zero because division by zero is undefined. Therefore, when determining the domain of rational functions, it's crucial to find the values of the variable that make the denominator zero and exclude them from the domain.
Rational functions can have asymptotes, that is, lines that the graph of the function approaches but never touches. In our examples, \( f(x) \) and \( g(x) \) could have vertical asymptotes at \( x = 4 \) and \( x = 6 \) respectively, representing points of non-definition in their domains.

Function Operations

Function operations involve adding, subtracting, multiplying, and dividing functions. When performing these operations, the domain of the resulting function is generally the intersection of the individual domains of the original functions.
For example, when finding \( f + g \) and \( f - g \), we take the domains of \( f(x) \) and \( g(x) \) and determine where they overlap. This means the resulting functions are undefined at both \( x = 4 \) and \( x = 6 \), so their domain in interval notation is \( (-\infty, 4) \cup (4, 6) \cup (6, \infty) \). The same applies to the product \( f \cdot g \) and quotient \( \frac{f}{g} \), as both require that neither of the original functions is undefined.

Undefined Expressions

Undefined expressions occur when mathematical operations do not result in a valid number, such as dividing by zero. In functions, ensuring these situations are avoided is key to determining the domain.
For rational functions like \( f(x) = \frac{1}{x-4} \), the expression becomes undefined when \( x = 4 \), as this would result in division by zero. Similarly, for \( g(x) = \frac{1}{6-x} \), it's undefined when \( x = 6 \).
When manipulating functions, such as taking their sum, difference, or quotient, you must ensure these undefined values are excluded from the domain. This careful approach aids in maintaining function validity across different operations.