Problem 11 - Beginning and Intermediate Algebra

Question

For each pair of functions, find a) \(\left(\frac{f}{g}\right)(x)\) and b \(\left(\frac{f}{g}\right)(-2)\) Identify any values that are not in the domain of \(\left(\frac{f}{g}\right)(x)\). $$f(x)=x^{2}-5 x-24, g(x)=x-8$$

Step-by-Step Solution

Answer

In conclusion: 1. \(\left(\frac{f}{g}\right)(x) = \frac{x^2 - 5x - 24}{x - 8}\) 2. \(\left(\frac{f}{g}\right)(-2) = 1\) 3. The domain of \(\left(\frac{f}{g}\right)(x)\) is all real numbers except \(x = 8\).

Step 1: Finding the function \(\left(\frac{f}{g}\right)(x)\)

The given functions are \(f(x) = x^2 - 5x - 24\) and \(g(x) = x - 8\). To find the function \(\left(\frac{f}{g}\right)(x)\), we will divide f(x) by g(x): \[\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 5x - 24}{x - 8}\] Now we have found the function \(\left(\frac{f}{g}\right)(x)\).

Step 2: Evaluating \(\left(\frac{f}{g}\right)(-2)\)

To find the value of \(\left(\frac{f}{g}\right)(-2)\), we will substitute x = -2 into the function \(\left(\frac{f}{g}\right)(x)\) obtained in step 1: \[\left(\frac{f}{g}\right)(-2) = \frac{(-2)^2 - 5(-2) - 24}{-2 - 8} = \frac{4 + 10 - 24}{-10} = \frac{-10}{-10} = 1\] So, \(\left(\frac{f}{g}\right)(-2) = 1\).

Step 3: Finding the domain of \(\left(\frac{f}{g}\right)(x)\)

The domain of \(\left(\frac{f}{g}\right)(x)\) is the set of all real values for x, except for the values that make the denominator equal to zero because division by zero is undefined. In this case, the denominator is \(g(x) = x - 8\). To find the values that make this function equal to zero, we solve the equation: \[x - 8 = 0\] Solving for x, we get \(x = 8\). Thus, the domain of \(\left(\frac{f}{g}\right)(x)\) is all real numbers except \(x = 8\). In conclusion: 1. \(\left(\frac{f}{g}\right)(x) = \frac{x^2 - 5x - 24}{x - 8}\) 2. \(\left(\frac{f}{g}\right)(-2) = 1\) 3. The domain of \(\left(\frac{f}{g}\right)(x)\) is all real numbers except \(x = 8\).

Key Concepts

Function Division made Simple

In algebra, dividing functions is similar to dividing numbers but done in terms of variable expressions. When we divide one function by another, it means we take a function like \( f(x) \) and divide it by another function \( g(x) \). This creates a new function, symbolically shown as \( \left( \frac{f}{g} \right)(x) \). For instance, if \( f(x) = x^2 - 5x - 24 \) and \( g(x) = x - 8 \), dividing them gives \( \left( \frac{f}{g} \right)(x) = \frac{x^2 - 5x - 24}{x - 8} \). Steps to divide functions:

• Identify the two functions involved in the division.
• Write the numerator (the function you are dividing) over the denominator (the function you are dividing by).
• Simplify the expression if possible, but remember it can remain as a fraction if simplification is not possible.

Function division is straightforward but requires careful handling, especially keeping an eye on the denominator that can affect the domain.

Domain of a Function Explained

The domain of a function is crucial since it tells you all possible input values (x-values) that a function can take without causing any mathematical errors like division by zero. For a function defined by division such as \( \left( \frac{f}{g} \right)(x) = \frac{x^2 - 5x - 24}{x - 8} \), the domain excludes any x-value that makes the denominator zero because division by zero isn't defined in math.To find the values not in the domain:

• Set the denominator equal to zero.
• Solve the equation to find the problematic x-value(s).
• Exclude those values from the domain.

In the example \( x - 8 = 0 \) leads to \( x = 8 \) indicating \( x = 8 \) is excluded from the domain of our function division. So, for this function, all real numbers are allowed except for x = 8. Always remember handling the domain carefully ensures correct and meaningful results.

Evaluate a Function with Confidence

Evaluating a function means finding the function's value for a particular input. Let's consider we have our function \( \left( \frac{f}{g} \right)(x) = \frac{x^2 - 5x - 24}{x - 8} \). We want to find \( \left( \frac{f}{g} \right)(-2) \). Steps to evaluate:

• Substitute the given value of x into the function wherever x appears.
• Simplify the expression step by step.
• Ensure all calculations are correct, consider negative signs and operations.

For this example, substitute -2 into the function's place of x: \( \frac{(-2)^2 - 5(-2) - 24}{-2 - 8} \). Calculate to get \( \frac{4 + 10 - 24}{-10} = \frac{-10}{-10} = 1 \), meaning \( \left( \frac{f}{g} \right)(-2) = 1 \). With practice and checking your work, evaluating a function becomes second nature.