Problem 22 - Algebra A Combined Function

Question

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=|x| ; g(x)=14 x-8 $$

Step-by-Step Solution

Answer

\((f \circ g)(x) = |14x - 8|\) and \((g \circ f)(x) = 14|x| - 8\).

Step 1: Understand the notation

The notation \((f \circ g)(x)\) represents the composition of functions \(f\) and \(g\), meaning \(f(g(x))\). The notation \((g \circ f)(x)\) represents \(g(f(x))\). Our task is to find these expressions.

Step 2: Evaluate \((f \circ g)(x)\)

To find \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\). So, \(g(x) = 14x - 8\), replace \(x\) in \(f(x) = |x|\) with \(14x - 8\). Thus, \((f \circ g)(x) = |14x - 8|\).

Step 3: Evaluate \((g \circ f)(x)\)

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). So \(f(x) = |x|\), replace \(x\) in \(g(x) = 14x - 8\) with \(|x|\). Thus, \((g \circ f)(x) = 14|x| - 8\).

Key Concepts

Absolute Value Function

The absolute value function is an essential concept in mathematics. It is represented by the notation \(|x|\), which gives the non-negative distance of a number \(x\) from zero on the number line. The absolute value function ensures the result is always non-negative, whether \(x\) is positive or negative. Let's break it down further with some characteristics:

• If \(x\) is positive, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\).
• The absolute value of zero is zero, i.e., \(|0| = 0\).

For example, if \(x = -3\), then \(|x| = |-3| = 3\). This concept is crucial when evaluating composite functions, ensuring any transformations occurring inside the absolute value function result in non-negative outputs only.

Algebraic Expressions

Algebraic expressions involve combining numbers, variables, and operations. Theyare foundational to understanding more complex mathematics like function composition.
An algebraic expression like \(14x - 8\) includes:

• Coefficients, such as 14, which multiply the variable \(x\).
• Constants, like -8, which are separate from the variable multiplication.
• Operations, such as addition and subtraction, guide how terms are combined.

When composing functions like \(f(x) = |x|\) and \(g(x) = 14x - 8\), you substitute one algebraic expression into another. This can modify the expression's structure and properties, allowing us to evaluate how changes in one function affect the other. Being able to manipulate and understand algebraic expressions is key to successfully evaluating function compositions.

Evaluation of Functions

In mathematics, evaluating functions involves substituting specific values into the given function, simplifying, and calculating the output. When dealing with function compositions, like \((f \circ g)(x)\) or \((g \circ f)(x)\), this process involves multiple steps:

• Identify the functions involved and their definitions. Here, \(f(x) = |x|\) and \(g(x) = 14x - 8\).
• Substitute one function into another. For \((f \circ g)(x)\), replace \(x\) in \(f(x)\) with the entire function \(g(x)\), which gives \(|14x - 8|\).
• Follow similar steps to evaluate \((g \circ f)(x)\), substituting \(f(x)\) into \(g(x)\), leading to \(14|x| - 8\).
• Simplify as necessary, focusing on ensuring your solution respects all Parts of the functions, like the absolute value component.

Understanding the evaluation process is essential for mastering function composition. It helps us observe how changes in the input of one function can directly influence the result of another.