Problem 79 - College Algebra

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

The composite function \((g \circ g)(x)\) is \(9x + 20\).

Step 1: Understand the Composite Function

The composite function \((g \circ g)(x)\) represents the function \(g(g(x))\). This means you substitute the function \(g(x)\) into itself, replacing \(x\) in the \(g(x)\) expression with \(g(x)\).

Step 2: Write the Expression for g(x)

Identify the function \(g(x)\). From the problem, \(g(x) = 3x + 5\). This will be used in the composition process.

Step 3: Substitute g(x) into g(x)

Substitute \(g(x)\) into \(g(x)\). Replace \(x\) in \(g(x) = 3x + 5\) with the entire expression \(g(x) = 3x + 5\).Thus, this becomes \(g(g(x)) = 3(3x + 5) + 5\).

Step 4: Simplify the Composite Function

Carry out the multiplication and addition in \(g(g(x)) = 3(3x + 5) + 5\).Calculate it as: - First, distribute the 3: \(3(3x) + 3(5) = 9x + 15\).- Then add 5: \(9x + 15 + 5 = 9x + 20\).

Step 5: Write the Final Expression

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

Key Concepts

Composite Functions

In mathematics, a composite function is essentially a function within another function. Imagine you have two functions, say \( f(x) \) and \( g(x) \). A composite function might look like \( (g \circ f)(x) \), which means you first apply \( f \) to \( x \) and then apply \( g \) to the result of \( f(x) \). In this specific exercise, we dealt with \((g \circ g)(x)\). What this means is applying \( g(x) \) to itself.

• You start by understanding what \( g(x) \) is, which is given as \( 3x + 5 \).
• Then you replace every \( x \) in \( g(x) \) with \( g(x) \) itself. This substitution is the key step in forming a composite function.

By comprehending composite functions, it becomes easier to see how functions can be combined to achieve complex expressions and results.

Function Operations

Function operations refer to ways that functions can interact or be manipulated together. These operations include addition, subtraction, multiplication, division, and composition of functions. Composition, as we saw with \((g \circ g)(x)\), is a specific type of operation that involves inserting one function into another.
When dealing with function operations, always:

• Understand the basic definition of each function involved.
• Plan how to combine these functions to reach the desired formation.

These operations not only help describe algebraic relationships but also reveal how different mathematical rules apply when two independent functions interact.
Function operations are an essential toolset for more advanced mathematical topics.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, and they often appear in the form \( ax^2 + bx + c \). In our problem, the quadratic function was \( f(x) = 2x^2 + 1 \), though we didn't directly use it in solving \((g \circ g)(x)\).
Some characteristics of quadratic functions include:

• When graphed, they form a parabola, either opening upwards or downwards depending on the sign of \( a \).
• The vertex represents the maximum or minimum point of the parabola.

Understanding how quadratic functions work is crucial, as they frequently arise in various mathematical contexts and applications, such as physics and engineering.

Linear Functions

The linear function in our exercise was \( g(x) = 3x + 5 \). Linear functions are first-degree polynomials, and their standard form is \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, where the line crosses the y-axis.

In this exercise, the linear function was used to form the composite function with itself, which involved substituting \( g(x) = 3x + 5 \) back into the same function.
Linear functions are straightforward but foundational to all algebra. They appear everywhere in problem-solving, modeling, and graph analysis. Understanding them enables you to see how changes in \( x \) affect \( y \) in a constant proportion.