Problem 15 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Composition

The given problem requires us to find \( \left(\frac{f}{g}\right)(-1) \). This means we need to evaluate the function \( \frac{f(x)}{g(x)} \) at \( x = -1 \). The function \( \frac{f}{g} \) simply represents the division of function \( f(x) \) by function \( g(x) \).

Step 2: Define the Functions

We have been given the functions \( f(x) = x^2 + 3x \) and \( g(x) = 2x - 1 \). These will help us create the function \( \frac{f}{g} \).

Step 3: Formulate the Division Function

To form the division function, we use: \[ \frac{f}{g}(x) = \frac{f(x)}{g(x)} = \frac{x^2 + 3x}{2x - 1} \].

Step 4: Substitute the Value of \(x = -1\)

Substitute \( x = -1 \) into the division function: \[ \frac{f}{g}(-1) = \frac{(-1)^2 + 3(-1)}{2(-1) - 1} \].

Step 5: Simplify the Numerator

Calculate the numerator: \((-1)^2 + 3(-1) = 1 - 3 = -2\).

Step 6: Simplify the Denominator

Calculate the denominator: \(2(-1) - 1 = -2 - 1 = -3\).

Step 7: Simplify the Fraction

Using the simplified numerator and denominator, we find: \(\frac{-2}{-3} = \frac{2}{3}\). Thus, \(\left(\frac{f}{g}\right)(-1) = \frac{2}{3}\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function for a specific input. Think of it like a machine where you put in a number, and it cranks out a result based on its internal rules or equation.

In our exercise, we are evaluating the function

• \( \frac{f}{g}(x) = \frac{f(x)}{g(x)} = \frac{x^2 + 3x}{2x - 1} \)

When evaluating at \( x = -1 \), we substitute this value directly into the function to get a numerical value. This involves replacing every occurrence of \( x \) with \( -1 \) in both the numerator \( f(x) \) and the denominator \( g(x) \).

By doing this, you will end up with a formula that can be simplified to provide the final result of the function at that specific input.

Rational Functions

Rational functions are expressions formed by dividing two polynomial functions. They have the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.

In this case, we have the rational function \( \frac{f(x)}{g(x)} = \frac{x^2 + 3x}{2x - 1} \). Here:

• The numerator, \( f(x) = x^2 + 3x \), is a polynomial of degree 2.
• The denominator, \( g(x) = 2x - 1 \), is a polynomial of degree 1.

A key aspect of rational functions is to consider the domain. The values of \( x \) for which the function is undefined are critical, particularly where the denominator equals zero. This serves as a reminder that rational functions can behave differently at these points.

Algebraic Simplification

Algebraic simplification is the process of reducing mathematical expressions into simpler or more manageable forms. This often involves combining like terms, factoring, and reducing fractions.

In our exercise, after substituting \( x = -1 \) into the rational function \( \frac{x^2 + 3x}{2x - 1} \), we get the fraction \( \frac{-2}{-3} \). The next step is simplifying \( \frac{-2}{-3} \) to \( \frac{2}{3} \).

Here, both the numerator and denominator are negative, and dividing two negative numbers results in a positive outcome. Therefore, despite the initial complexity, the operation results in a straightforward positive fraction.

Simplification is crucial as it provides a clear and concise representation of the solution, making it more understandable and applicable.