Problem 57 - Precalculus Essentials

Question

Find. a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=x^{2}+2, g(x)=x^{2}-2$$

Step-by-Step Solution

Answer

The expressions for (f ∘ g)(x) and (g ∘ f)(x) are \(x^{4} - 4x^{2} + 6\) and \(x^{4} + 4x^{2} + 2\), respectively. The values of (f ∘ g)(2) and (g ∘ f)(2) are 6 and 22, respectively.

Step 1: Compute (f ∘ g) (x)

The composition \(f \circ g\)(x) means applying function g to x, then applying function f to the result. We start by finding g(x), which is \(x^{2} - 2\). Substitution of this into function f yeilds \(f(g(x)) = (x^{2} - 2)^{2} + 2 = x^{4} - 4x^{2} + 6 \).

Step 2: Compute (g ∘ f) (x)

The composition \(g \circ f\)(x) means applying function f to x, then applying function g to the result. We start by finding f(x), which is \(x^{2} + 2\). Substitution of this into function g yields \(g(f(x)) = (x^{2} + 2)^{2} - 2 = x^{4} + 4x^{2} + 2 \).

Step 3: Compute (f ∘ g) (2)

To compute the value of the composition at x = 2, we substitute x = 2 in the function we obtained in Step 1. So \(f(g(2)) = (2^{4} - 4*2^{2} + 6 = 6\).

Step 4: Compute (g ∘ f) (2)

Similarly, to compute the value of the composition at x = 2, we substitute x = 2 in the function we obtained in Step 2. So \(g(f(2)) = (2^{4} + 4*2^{2} + 2 = 22 \).

Key Concepts

Polynomial Functions

Polynomial functions are a type of algebraic expression that consists of variables and coefficients organized in terms. Each term is composed of a coefficient and a variable raised to a non-negative integer exponent.
A general polynomial function is expressed as:

• Each term looks like this: \(a_nx^n\), where \(a_n\) is a coefficient and \(n\) is a non-negative integer representing the degree of the term.
• The full polynomial function can be written as \(a_nx^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_1, a_0\) are coefficients.

Polynomial functions are continuous and smooth, making them ideal models for a variety of real-world phenomena. In our exercise, we deal with simple polynomials: \(f(x) = x^2 + 2\) and \(g(x) = x^2 - 2\). Each of these polynomials is a quadratic, characterized by the degree of \(2\), indicating they have a parabolic shape.

Substitution Method

The substitution method is a common technique when working with functions, especially in the context of function compositions.
The idea is to "substitute," or replace, one part of an expression with another.
Let's understand the steps involved:

• First, compute the innermost function. For example, in \(f(g(x))\), find \(g(x)\).
• Next, substitute the expression found from the first function into the outer function. For \(f(g(x))\), plug the result from \(g(x)\) into \(f\).
• Perform any necessary algebraic simplifications to get the final resultant expression.

In our exercise, for \((f \circ g)(x)\), we replaced \(x\) in \(f(x) = x^2 + 2\) with \(g(x) = x^2 - 2\) and simplified: \((x^2 - 2)^2 + 2\).
Similarly, for \((g \circ f)(x)\), substitute \(f(x) = x^2 + 2\) into \(g(x) = x^2 - 2\) and simplify the expression.

Function Evaluation

Function evaluation is the process of finding the value of a function at a specific point.
To evaluate a function, we substitute the given point into the function's expression and simplify.
Here are the key aspects of function evaluation:

• Identify the function you want to evaluate. This could be a standalone function like \(f(x)\) or a composite function such as \((f \circ g)(x)\).
• Substitute the point into the function. If evaluating at \(x = 2\), replace every \(x\) in the expression with \(2\).
• Simplify the expression to obtain the result. This means computing any exponents, multiplying, and adding or subtracting as needed.

In our problem, after determining the expression for \((f \circ g)(x)\) and \((g \circ f)(x)\), we substituted \(x = 2\) into each composite to find their values:

• \((f \circ g)(2) = 6\)
• \((g \circ f)(2) = 22\)

Function evaluation is crucial for drawing insights into how functions behave at specific points.