Problem 26 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find \(f(x)\) for the indicated values of \(x\), if possible. (b) Find the domain of \(f\). $$ f(x)=\frac{3 x-5}{x+5} \text { for } x=-1, a $$

Step-by-Step Solution

Answer

(a) \(f(-1) = -2\), \(f(a) = \frac{3a-5}{a+5}\). (b) Domain: \((-\infty, -5) \cup (-5, \infty)\).

Step 1: Find f(-1)

To calculate \(f(-1)\), we substitute \(-1\) for \(x\) in the function. \[ f(-1)=\frac{3(-1)-5}{-1+5}=\frac{-3-5}{4}=\frac{-8}{4}=-2 \] Thus, \(f(-1)=-2\).

Step 2: Substitute 'a' into f(x)

Next, we substitute \(a\) for \(x\) to find \(f(a)\). \[ f(a)=\frac{3a-5}{a+5} \] This expression represents \(f(a)\).

Step 3: Determine the Domain of f(x)

The domain of a function is the set of all possible input values (\(x\)) for which the function is defined. Since the function \(f(x)=\frac{3x-5}{x+5}\) has a denominator, the function is undefined whenever the denominator is zero. To find when the denominator is zero, we solve: \[ x+5=0 \] Solving this gives \(x=-5\). Therefore, the domain excludes \(x=-5\) and is all real numbers except \(-5\). In interval notation, the domain is: \( (-\infty, -5) \cup (-5, \infty) \).

Key Concepts

Rational Functions

Rational functions are a type of function defined by the ratio of two polynomials. In simpler terms, it is a fraction where the numerator and the denominator are both polynomials. The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.
What makes rational functions interesting is how the value of \( x \) can affect their definition. The function is only valid when the denominator \( Q(x) \) does not equal zero, because division by zero is undefined in mathematics.
This idea is key in determining the domain of the rational function.
Rational functions often exhibit asymptotic behavior, which means as \( x \) approaches certain values, the function either shoots up to infinity or down to negative infinity. This behavior typically occurs near values that make the denominator zero, making these spots crucial when analyzing the graph or the properties of the function.

Interval Notation

Interval notation is a concise way to express a range of numbers, especially when describing the domain or range of a function. Instead of writing a long list of numbers or using inequalities, interval notation captures everything succinctly.
There are a few symbols to be aware of when using interval notation:

• "(" or ")" means that the endpoint is not included in the interval, often called open intervals.
• "[" or "]" means the endpoint is included, known as closed intervals.
• The symbol \(\cup\) is used to join disjoint intervals.

When describing the domain of the rational function in the exercise, interval notation helps specify all valid \( x \) values efficiently. For instance, \( (-\infty, -5) \cup (-5, \infty) \) encompasses all real numbers except \( x = -5 \). It shows that \( x \) can take any value from negative infinity up to \(-5\) without including it and also from just above \(-5\) to infinity. This is particularly useful when communicating which inputs are valid for a function that has certain restrictions.

Undefined Expressions

Undefined expressions in math occur under specific circumstances, such as when a denominator equals zero in a fraction. This is because dividing by zero does not produce a finite number or result. In the context of functions, an undefined expression means that for certain values of \( x \), the function does not yield a real number as an output.
Understanding where a function is undefined is crucial, especially for rational functions. It directly relates to the domain of the function, as these identified values need to be excluded. For the function \( f(x) = \frac{3x-5}{x+5} \), substitute \( x + 5 = 0 \) to find when it becomes undefined. Solving this equation, we find \( x = -5 \).
These situations are more than just mathematical rules; they have practical purposes in ensuring functions behave predictively and don't lead to paradoxical or infinite results. Recognizing where expressions are undefined helps avoid errors both in calculations and in understanding the behavior of the function.