Problem 20 - 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. $$\left(\frac{f}{g}\right)\left(\frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The expression is undefined because the denominator is zero.

Step 1: Understand the Composite Function \(\frac{f}{g}(x)\)

The goal is to find \(\frac{f}{g}(x)\), which means taking the function \(f(x)\) and dividing it by \(g(x)\). Mathematically, this is written as \(\frac{f(x)}{g(x)}\).

Step 2: Substitute the Given Functions into \(\frac{f}{g}(x)\)

Substitute \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\) into \(\frac{f}{g}(x)\). This results in \(\frac{x^2 + 3x}{2x - 1}\).

Step 3: Evaluate \(\frac{f}{g}\left(\frac{1}{2}\right)\)

To evaluate \(\frac{f}{g}\left(\frac{1}{2}\right)\), substitute \(x = \frac{1}{2}\) into the expression \(\frac{x^2 + 3x}{2x - 1}\). Calculate the numerator: \(\left(\frac{1}{2}\right)^2 + 3\left(\frac{1}{2}\right) = \frac{1}{4} + \frac{3}{2}\). For the denominator, calculate \(2\left(\frac{1}{2}\right) - 1\).

Step 4: Calculate the Numerator

Calculate the numerator: \(\frac{1}{4} + \frac{3}{2}\) can be rewritten as \(\frac{1}{4} + \frac{6}{4} = \frac{7}{4}\).

Step 5: Calculate the Denominator

Calculate the denominator: \(2\left(\frac{1}{2}\right) - 1 = 1 - 1 = 0\).

Step 6: Determine the Final Result

Since the denominator is zero, the expression \(\frac{f}{g}\left(\frac{1}{2}\right) = \frac{\frac{7}{4}}{0}\) is undefined.

Key Concepts

Composite Function

A composite function is an operation where one function is applied inside another. In the specified exercise, instead of combining functions with addition or multiplication, we employ division. Here, two functions, \( f(x) = x^2 + 3x \) and \( g(x) = 2x - 1 \), are combined in such a way that the output of one function becomes the input for the dividing operation.

• Composite function emphasizes the sequence in which functions are applied.
• It requires a clear understanding of function properties to avoid undefined operations, especially with division.

This concept is crucial as it opens up advanced topics in calculus and real-world applications, especially in scenarios such as calculating rates and accumulations.

Algebraic Expressions

Algebraic expressions involve a combination of constants, variables, and arithmetic operators. In this exercise, the expressions \( f(x) = x^2 + 3x \) and \( g(x) = 2x - 1 \) demonstrate how algebra can be used to build complex relationships and operations.

• Each expression consists of terms and coefficients. For example, \( x^2 \) is a term with a coefficient of 1, while \( 3x \) has a coefficient of 3.
• Understanding the structure of these expressions helps in manipulating and simplifying functions during operations like addition, subtraction, multiplication, and division.

Recognizing and manipulating algebraic expressions effectively is an essential skill in navigating mathematical challenges since it allows for solving equations and understanding function behavior.

Mathematical Operations

Mathematical operations provide the basic tactics to manipulate and evaluate functions. The operation of division in this exercise, represented by \( \left( \frac{f}{g} \right)(x) \), involves dividing the output of one function by the output of another. To execute the division here:

• First, calculate the expressions \( f(x) \) and \( g(x) \) independently for a given \( x \).
• Then, divide their results. It is crucial to manage zeros in the denominator to avoid undefined results. As shown, even sophisticated operations can fail when division leads to results like \( \frac{7}{4} \) divided by 0, rendering the expression undefined.

Mastering mathematical operations is paramount for succeeding in more complex fields, as they form the foundation upon which advanced topics are built. Mathematics is all about accuracy, and operations guide us through the meticulous process of transformation and evaluation.