Problem 79 - College Algebra with Corequisite Support

Question

For the following exercises, use the functions \(f(x)=2 x^{2}+1\) and \(g(x)=3 x+5\) to evaluate or find the composite function as indicated. \((g \circ g)(x)\)

Step-by-Step Solution

Answer

(g ∘ g)(x) = 9x + 20.

Step 1: Understanding the Definition of Composite Functions

The notation \((g \circ g)(x)\) represents the composition of the function \(g(x)\) with itself. This means we substitute \(g(x)\) into itself, resulting in \(g(g(x))\).

Step 2: Substitute g(x) into Itself

Start by substituting \(g(x) = 3x + 5\) into itself, which means replacing every \(x\) in \(g(x)\) with \(3x + 5\). Thus, \(g(g(x)) = g(3x + 5)\).

Step 3: Simplify the Expression

Replace \(x\) in \(g(x) = 3x + 5\) with \(3x + 5\). This gives: \[\begin{align*}g(g(x)) &= 3(3x + 5) + 5 \&= 9x + 15 + 5 \&= 9x + 20.\end{align*}\]

Step 4: Write the Final Result

The final expression for \((g \circ g)(x)\) after simplifying is \(9x + 20\).

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics, particularly in algebra. It involves combining two functions in such a way that the output of one function becomes the input of another. This is symbolically represented as

• \((f \circ g)(x) = f(g(x))\)

In this context, composition operates much like a function factory line, where function \(g(x)\) processes an input, and its result is passed through function \(f(x)\) for further transformation. In the exercise provided,

• \((g \circ g)(x)\) means that the function \(g(x)\) is applied to itself, or \(g(g(x))\).

This results in the nested composition of \(g(x)\), creating a new expression which combines two layers of the function \(g(x)\), altering the original input \(x\) more significantly than a single function would.

Algebraic Functions

Algebraic functions are mathematical expressions involving variables, numbers, and operations such as addition, subtraction, multiplication, division, and taking roots. These functions can take various forms, from simple linear functions to more complex polynomial expressions.The functions in the original exercise,

• \(f(x) = 2x^2 + 1\) and
• \(g(x) = 3x + 5\)

are examples of algebraic functions.Function \(f(x)\) is a quadratic function because it includes an \(x^2\) term, giving it a parabolic shape on a graph. The function \(g(x)\) is a linear function, characterized by its constant rate of change and straight-line graph. By substituting one function into another, which is common in function composition, we create a new algebraic function with potentially different characteristics from the original functions.

Function Evaluation

Function evaluation is the process of finding the output of a function given an input. This is done by replacing the variable in the function's equation with the given value or expression. In the exercise, we evaluated the composite function

• \((g \circ g)(x)\)

by substituting \(g(x)\) into itself, which required placing \(3x+5\) in the function's position of \(x\). This creates a double layer of substitution, meaning every \(x\) in the function \(g(x)\) was replaced with \((3x + 5)\), leading to the result

• \[g(g(x)) = 3(3x + 5) + 5 = 9x + 20\]

Function evaluation helps simplify and find specific values or new functions based on operated inputs, such as in composition cases. It is an integral skill in solving algebraic problems and understanding how complex function interactions work.