Problem 21 - Algebra 2 and Trigonometry

Question

\(\operatorname{In} 19-22,\) let \(\mathrm{f}(x)=|x| \cdot\) Find \(\mathrm{f}(\mathrm{g}(x))\) and \(\mathrm{g}(\mathrm{f}(x))\) for each given function. $$ g(x)=2 x+3 $$

Step-by-Step Solution

Answer

\( f(g(x)) = |2x + 3| \) and \( g(f(x)) = 2|x| + 3 \).

Step 1: Understanding the Functions

First, we need to understand the functions involved. The function \( f(x) = |x| \) represents the absolute value of \( x \). The function \( g(x) = 2x + 3 \) is a linear function that transforms \( x \) by scaling it by 2 and then translating it by 3.

Step 2: Find \( f(g(x)) \)

To find \( f(g(x)) \), substitute \( g(x) = 2x + 3 \) into \( f(x) = |x| \). Thus, \( f(g(x)) = |2x + 3| \). This means we take the absolute value of the expression \( 2x + 3 \).

Step 3: Simplify \( f(g(x)) \)

The expression \( f(g(x)) = |2x + 3| \) is simplified by understanding that it just means the magnitude of the value \( 2x + 3 \); there are no further simplifications in the general form without specific \( x \) values.

Step 4: Find \( g(f(x)) \)

Now, find \( g(f(x)) \) by substituting \( f(x) = |x| \) into \( g(x) = 2x + 3 \). So, \( g(f(x)) = g(|x|) = 2|x| + 3 \). This modifies \( |x| \) by scaling it by 2 and then translating it by 3.

Step 5: Simplify \( g(f(x)) \)

The expression \( g(f(x)) = 2|x| + 3 \) is already in its simplest form. It represents a linear transformation of the magnitude of \( x \).

Key Concepts

Absolute Value Function

The absolute value function is a simple yet powerful concept in algebra. It is typically denoted as \(|x|\), representing the distance of a number \(x\) from zero on the number line. This function always outputs non-negative values, regardless of whether \(x\) is positive or negative.

• If \(x\) is positive or zero, the absolute value simply equals \(x\).
• If \(x\) is negative, the absolute value is \(-x\). This turns the negative value into a positive one.

The absolute value has many practical applications in mathematics, from solving equations to measuring distances. In the context of the exercise, when you substitute a linear function like \(g(x) = 2x + 3\) into the absolute value function \(f(x) = |x|\), you derive \(f(g(x)) = |2x + 3|\). Here, it’s crucial to realize that the result is a reflection of the function’s behavior that ignores the sign of \(2x + 3\), thus always creating a non-negative result.

Linear Function

Linear functions are one of the most straightforward forms of algebraic functions you will encounter. They are characterized by a polynomial of degree one, meaning their graph is a straight line. The general form of a linear function is \(g(x) = mx + b\), where \(m\) is the slope, dictating the steepness of the line, and \(b\) is the y-intercept, where it crosses the y-axis.

In the given exercise, the linear function \(g(x) = 2x + 3\) modifies \(x\) by doubling it and then adding 3. This transformation changes the input value, shifting it both vertically and horizontally on the graph. When we compose this linear function with the absolute value function, as in \(g(f(x)) = 2|x| + 3\), the linear function operates on the non-negative outputs of the absolute value function. This builds a combination where the final outcome is determined through both the shape of the absolute value graph and the line generated by the linear transformation.

Algebraic Functions

Algebraic functions cover a broad category in mathematics that includes polynomial, rational, and radical functions. These functions are defined by algebraic expressions, using operations such as addition, subtraction, multiplication, division, and root extraction.

In the exercise, the composition of functions \(f\) and \(g\) creates expressions like \(f(g(x)) = |2x + 3|\) and \(g(f(x)) = 2|x| + 3\). Such compositions illustrate the power of algebraic functions in simplifying complex tasks. Composing algebraic functions often requires replacing one function within another, forming a new dependent relationship on the input value.

• \(f(g(x))\): This combines the absolute nature of \(f(x)\) with the linear transformation of \(g(x)\), indicating transformations in both magnitude and direction.
• \(g(f(x))\): Demonstrates how outputs from one function serve as inputs for another, advancing the result through sequential functional steps.

Understanding these concepts enriches your algebra toolkit, helping solve numerous mathematical challenges.