Problem 32 - Intermediate Algebra

Question

Let \(f(x)=2 x-5\) and \(g(x)=x+1 .\) Find each of the following function values. See Example 2 . $$ (f / g)(2) $$

Step-by-Step Solution

Answer

\((f/g)(2) = \frac{-1}{3}\).

Step 1: Understand the Function Composition

Given two functions, \(f(x) = 2x - 5\) and \(g(x) = x + 1\), we need to find the value of \((f/g)(2)\). This represents the division of the function \(f(x)\) by \(g(x)\) evaluated at \(x = 2\). This means calculating \(\frac{f(2)}{g(2)}\).

Step 2: Evaluate f(x) at x = 2

Substitute \(x = 2\) into the function \(f(x) = 2x - 5\):\[ f(2) = 2(2) - 5 = 4 - 5 = -1 \]

Step 3: Evaluate g(x) at x = 2

Substitute \(x = 2\) into the function \(g(x) = x + 1\):\[ g(2) = 2 + 1 = 3 \]

Step 4: Divide f(2) by g(2)

Now, divide the values found in the previous steps:\[ \frac{f(2)}{g(2)} = \frac{-1}{3} \]Thus, \((f/g)(2) = \frac{-1}{3}\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function for a specific input. In our case, we have two functions: \(f(x) = 2x - 5\) and \(g(x) = x + 1\). To evaluate a function, simply replace the variable \(x\) with the specified number and solve the expression.

• For \(f(x)\), when \(x = 2\): Substitute 2 into the function: \(f(2) = 2(2) - 5\).
• For \(g(x)\), when \(x = 2\): Substitute 2 into the function: \(g(2) = 2 + 1\).

By evaluating these functions, you determine the output based on the input, which is foundational in understanding how these functions behave with different variables.

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. While it's not directly used in this exercise, understanding function composition is crucial for complex function manipulation. To compose functions \(f\) and \(g\), you'd typically replace \(g(x)\) into \(f(x)\), or vice versa. But here, we focus on function division, not composition. Here's a quick example of function composition:

• If \(f(x) = 2x - 5\) and \(g(x) = x + 1\), then \(f(g(x)) = 2(x + 1) - 5\).
• Calculate it by replacing \(x\) in \(f\) with \(g(x)\). This gives us: \(2(x + 1) - 5 = 2x + 2 - 5 = 2x - 3\).

Even though this example doesn't directly apply to our problem, mastering function composition enhances overall mathematical fluency.

Rational Functions

A rational function is essentially a fraction where the numerator and/or denominator are polynomials. In this exercise, we consider the division of two functions, which creates a simple rational function. In the problem, the rational function is expressed as \((f/g)(x) = \frac{f(x)}{g(x)}\). At \(x = 2\), you substitute the evaluated values of \(f\) and \(g\):

• Numerator: \(f(2) = -1\)
• Denominator: \(g(2) = 3\)

Thus, \((f/g)(2) = \frac{-1}{3}\). Understanding rational functions is essential because they frequently appear in mathematical computations. Always remember that a rational function is undefined if its denominator is zero, but in our case, \(g(2) = 3\) so the division is safe.