Problem 31 - 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 \(f\). $$ f(x)=\frac{1}{x^{2}-9} \text { for } x=4, a-5 $$

Step-by-Step Solution

Answer

(a) \(f(4) = \frac{1}{7}\), \(f(a-5) = \frac{1}{a^2 - 10a + 16}\). (b) Domain: all \(x\) except \(x=3\) and \(x=-3\).

Step 1: Evaluate f(x) at x=4

To find \(f(4)\), substitute \(x = 4\) into the function: \[ f(4) = \frac{1}{4^2 - 9} = \frac{1}{16 - 9} = \frac{1}{7} \]. So, \(f(4) = \frac{1}{7}\).

Step 2: Evaluate f(x) at x=a-5

To find \(f(a-5)\), substitute \(x = a-5\) into the function: \[ f(a-5) = \frac{1}{(a-5)^2 - 9} \]. This expression simplifies to \[ f(a-5) = \frac{1}{a^2 - 10a + 25 - 9} = \frac{1}{a^2 - 10a + 16} \]. So, \(f(a-5)\) is defined as long as \(a^2 - 10a + 16 eq 0\).

Step 3: Determine the domain of f

The function \(f(x)\) is defined for all values of \(x\) except when the denominator is zero. Set the denominator equal to zero and solve: \[ x^2 - 9 = 0 \]. Factoring gives \((x-3)(x+3) = 0\), so \(x = 3\) or \(x = -3\). Thus, the domain of \(f\) is all real numbers \(x\) except \(x = 3\) or \(x = -3\).

Key Concepts

Evaluating functions

Evaluating a function means finding its value for specific inputs. Here's how you can tackle this process effectively:

• Select the function: Begin with identifying the function you need to evaluate. For example, for the function \( f(x) = \frac{1}{x^2 - 9} \).
• Substitute the input: Replace the variable \(x\) with the given values. For example, substitute \(x\) with 4 in \( f(x) = \frac{1}{x^2 - 9} \) to get \( f(4) = \frac{1}{4^2 - 9} \).
• Simplify the expression: Carry out the arithmetic to simplify, like \( f(4) = \frac{1}{16 - 9} = \frac{1}{7} \).

Evaluating a function helps us understand how the function behaves with certain inputs. Always ensure the function is defined for the inputs you are using—some inputs may cause undefined results, such as division by zero.

Rational functions

Rational functions involve ratios of polynomials. They often appear in the format \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. Here are key features of rational functions to remember:

• Domain concerns: Rational functions are not defined when the denominator, \(Q(x)\), equals zero. To find restricted values, solve \(Q(x) = 0\).
• Characteristics: The behavior of rational functions can include vertical asymptotes where the function is undefined. For example, in \( f(x) = \frac{1}{x^2 - 9} \), the vertical asymptotes occur at \(x = 3\) and \(x = -3\).

Understanding rational functions is crucial because they model many real-world phenomena, such as rates, proportions, and variations. Pay attention to any restrictions on the variable to ensure accurate evaluations and conclusions.

Function notation

Function notation is a streamlined way to express functions and handle variables. Here's a quick breakdown of its use:

• Symbol representation: Functions are typically represented as \( f(x) \), which defines a rule mapping \(x\) to an output. The letter can vary (e.g., \(g(x), h(x)\)).
• Input-output relationship: \( f(x) \) signifies the value of the function for the input \(x\). For \( f(x) = \frac{1}{x^2 - 9} \), the mathematical operation performed on \(x\) is set by the function.
• Convenience and clarity: Using function notation makes it easier to work with varying inputs and see clearly how a function changes with different arguments.

Function notation simplifies the process of writing and manipulating functions. It's a universal language in math that ensures everyone is on the same page when discussing how inputs transform into outputs.