Problem 17 - Algebra A Combined Function

Question

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=2 x-3 ; g(x)=x+7 $$

Step-by-Step Solution

Answer

\((f \circ g)(x) = 2x + 11\) and \((g \circ f)(x) = 2x + 4\).

Step 1: Understand Composite Functions

A composite function combines two functions into one, applying one function to the result of the other. For (f \circ g)(x), you first apply \(g(x)\) and then \(f(x)\). For (g \circ f)(x), the process is reversed.

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

First, identify \(g(x) = x + 7\). Replace \(x\) in \(f(x) = 2x - 3\) with \(g(x)\):\[(f \circ g)(x) = f(g(x)) = f(x+7) = 2(x + 7) - 3.\]Simplify the expression:\[= 2x + 14 - 3= 2x + 11.\]

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

First, identify \(f(x) = 2x - 3\). Replace \(x\) in \(g(x) = x + 7\) with \(f(x)\):\[(g \circ f)(x) = g(f(x)) = g(2x - 3) = (2x - 3) + 7.\]Simplify the expression:\[= 2x - 3 + 7= 2x + 4.\]

Key Concepts

Function Composition

Understanding composite functions is a vital step in grasping more complex algebraic concepts. A composite function blends two functions, where the output of one function becomes the input of the other. This interaction is denoted by the symbol \(\circ\). For instance, \((f \circ g)(x)\) means applying the function \(g(x)\) first, followed by \(f(x)\). In contrast, \((g \circ f)(x)\) involves applying \(f(x)\) first and then \(g(x)\).

• To compute \((f \circ g)(x)\): start with \(g(x)\), and plug the result into \(f(x)\).
• To calculate \((g \circ f)(x)\): begin with \(f(x)\), and use its output as the input for \(g(x)\).

This method underlines the dependent relationship between two functions and shows how they interrelate to process inputs into outputs.

Algebraic Functions

Algebraic functions involve expressions created using polynomial terms and operations such as addition, subtraction, multiplication, division, and root extraction. In the context of function composition, algebraic functions provide a structured way to express how one function acts upon another.
For example, given \(f(x) = 2x - 3\) and \(g(x) = x + 7\), calculating \((f \circ g)(x)\) involves substituting the expression for \(g(x)\) into \(f(x)\). This process requires arithmetic simplifications to arrive at the finished form of the composite function:

• \((f \circ g)(x) = 2(x + 7) - 3 = 2x + 11\)
• \((g \circ f)(x) = (2x - 3) + 7 = 2x + 4\)

Dealing with algebraic expressions involves consistent practice and familiarity with combining terms to arrive at the simplest form of an equation.

Function Notation

Function notation is a simplified and efficient way to refer to algebraic operations and relationships between variables. It uses a defined symbol, usually expressed as \(f(x)\), to denote the function's rule and how it works on variable \(x\). This notation not only represents but also clarifies the relationship between inputs and outputs for given functions.

• \(f(x)\) refers to a particular function, such as \(f(x) = 2x - 3\).
• \(g(x)\) might represent another, like \(g(x) = x + 7\).

Using function notation allows us to systematically analyze and solve problems involving composite functions, since it clearly shows which operations are to be performed and in what order. It simplifies expressing how one function acts upon another and illustrates the steps to find results such as \((f \circ g)(x)\) or \((g \circ f)(x)\). Understanding function notation thus plays a crucial role in simplifying complex mathematical processes.