Problem 18 - Algebra A Combined Function

Question

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=x+10 ; g(x)=3 x+1 $$

Step-by-Step Solution

Answer

\\((f \, \circ \, g)(x) = 3x + 11\\); \\(g \, \circ \, f)(x) = 3x + 31\\).

Step 1: Understand Composite Functions

A composite function \(f \, \circ \, g\)(x) means that you substitute the function \g(x)\ into every \x\ in the function \f(x)\. Similarly, \(g \, \circ \, f\)(x) means substituting \f(x)\ into every \x\ in the function \g(x)\.

Step 2: Compute f ∘ g(x)

To find \(f \, \circ \, g\)(x), substitute \g(x) = 3x + 1\ into \f(x) = x + 10\: \\(f \, \circ \, g\)(x) = f(g(x)) = f(3x + 1) = (3x + 1) + 10\. Simplify the expression: \(3x + 1 + 10 = 3x + 11\). So, \(f \, \circ \, g\)(x) = 3x + 11\.

Step 3: Compute g ∘ f(x)

To find \(g \, \circ \, f\)(x), substitute \f(x) = x + 10\ into \g(x) = 3x + 1\: \\(g \, \circ \, f\)(x) = g(f(x)) = g(x + 10) = 3(x + 10) + 1\. Distribute and simplify: \3(x + 10) + 1 = 3x + 30 + 1 = 3x + 31\. So, \(g \, \circ \, f\)(x) = 3x + 31\.

Key Concepts

Function Composition

In mathematics, function composition is a simple yet powerful concept. It involves creating a new function by applying one function to the result of another. Imagine you have two functions, say \( f \) and \( g \). Function composition allows us to combine them into a third function, which we'll denote as \( (f \circ g)(x) \). This notation implies that we first apply \( g \) to \( x \), and then apply \( f \) to the result of \( g(x) \). Similarly, \( (g \circ f)(x) \) means we first apply \( f \) and then \( g \).

• \( (f \circ g)(x) \): Apply function \( g \), then \( f \) to the outcome.
• \( (g \circ f)(x) \): Apply function \( f \), then \( g \) to the outcome.

The order of application matters in function composition, meaning \( (f \circ g)(x) \) is not necessarily the same as \( (g \circ f)(x) \). It's a crucial property that helps to create transformation sequences in various mathematical problems.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In our original problem, we dealt with algebraic expressions like \( f(x) = x + 10 \) and \( g(x) = 3x + 1 \). These expressions describe two different linear functions.

• Variables: Typically represented by letters like \( x \), allowing the expression to adopt different values.
• Constants: Fixed numbers such as the '+10' in \( f(x) = x + 10 \) or '+1' in \( g(x) = 3x + 1\).
• Operations: Actions such as addition, subtraction, multiplication, or division used to combine variables and constants.

Understanding these elements in algebraic expressions helps us determine how to manipulate and interpret functions, allowing function composition to be executed effectively. Simplifying and rewriting these expressions is critical in solving complex algebraic problems.

Mathematical Operations

Mathematical operations are the processes we use to combine numbers and variables, and they're foundational to creating and solving algebraic expressions. The operations used in our problem include both addition and multiplication. Here's how they work in function composition:

• Addition: In expressions like \( f(x) = x + 10 \) and \( g(x) = 3x + 1 \), addition combines constants with variables.
• Multiplication: When we compute \( g(f(x)) \), we multiply \( 3 \) by the entire expression \( (x + 10) \), making it \( 3(x + 10) \).

The order and type of operations matter significantly. If applied in different orders or ways, it affects the outcome of the expression. Simplifying involves carrying out these operations and ensuring each step respects mathematical principles and constraints. By performing operations in serving their specific role correctly, we unlock the power behind expressions and their compositions.