Problem 16 - A Graphical Approach to College Algebra

Question

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$\left(\frac{f}{g}\right)(4)$$

Step-by-Step Solution

Answer

\( \left( \frac{f}{g} \right)(4) = 4 \)

Step 1: Understand the Composition

The function notation \( \left( \frac{f}{g} \right)(x) \) means \( \frac{f(x)}{g(x)} \). Therefore, to find \( \left( \frac{f}{g} \right)(4) \), we must first evaluate \( f(4) \) and \( g(4) \).

Step 2: Evaluate \( f(4) \)

Use the function \( f(x) = x^2 + 3x \). Substitute \( x = 4 \) into this function:\[ f(4) = 4^2 + 3 \times 4 = 16 + 12 = 28 \]

Step 3: Evaluate \( g(4) \)

Use the function \( g(x) = 2x - 1 \). Substitute \( x = 4 \) into this function:\[ g(4) = 2 \times 4 - 1 = 8 - 1 = 7 \]

Step 4: Compute \( \left( \frac{f}{g} \right)(4) \)

Now form the fraction using the results from Steps 2 and 3:\[ \left( \frac{f}{g} \right)(4) = \frac{f(4)}{g(4)} = \frac{28}{7} = 4 \]

Key Concepts

Function Evaluation

When you're working with functions, one of the key skills to master is function evaluation. It's a process of finding the output of a function for a given input. Think of functions as machines, where you input a number (or 'x' value) and get an output. This output is determined by the function's formula.

In the given exercise, we have two functions to evaluate: \( f(x) = x^2 + 3x \) and \( g(x) = 2x - 1 \). To evaluate these functions at a specific point, like \( x = 4 \), substitute the number into the function in place of \( x \).

Let's break it down:

• For \( f(x) \), substitute 4 for \( x \) to get \( f(4) = 4^2 + 3 \times 4 \).
• For \( g(x) \), substitute 4 to get \( g(4) = 2 \times 4 - 1 \).

Think of this like replacing a variable with a number. It's as simple as that! This skill is essential for solving problems involving functions and helps build a strong math foundation.

Algebraic Fractions

Algebraic fractions involve fractions where the numerator, the denominator, or both are algebraic expressions. Solving problems with algebraic fractions requires specific steps, often involving multiple functions. In our exercise, we dealt with a composition of two functions, making it an algebraic fraction.

Here's how it works: the composition \( \left( \frac{f}{g} \right)(x) \) represents the division of two functions, creating a fraction with \( f(x) \) as the numerator and \( g(x) \) as the denominator.
This approach is essential for understanding how complex relationships between functions can be expressed mathematically.

• First, evaluate both functions individually at a certain point.
• Next, plug these values into the fraction \( \frac{f(x)}{g(x)} \).
• Finally, simplify the fraction to obtain the result.

The goal of working with algebraic fractions like this is to carefully handle expressions to maintain accuracy, especially when dealing with compositions of functions.

Problem Solving

Problem solving with functions often involves multiple steps to find a solution. It's like solving a puzzle where each part must fit in a specific order. Let's explore how we can approach such problems effectively.

In our exercise, we handled a composition of functions and used a variety of algebraic skills:

• Identify and understand the required operation or composition, such as \( \left( \frac{f}{g} \right)(x) \).
• Break down the main problem into simpler parts: evaluate the functions separately and then combine using algebra.
• Ensure each step is clear by writing it down, checking calculations, and simplifying the results.

By approaching the problem methodically, as we did: find \( f(4) \), find \( g(4) \), and then \( \frac{28}{7} \), you can solve complex problems more easily. Practicing this way enhances critical thinking and analytical skills, paving the way for tackling more advanced mathematical concepts.