Problem 45 - Calculus Volume 1

Question

For the following exercises, for each pair of functions, find a. \((f \circ g)(x)\) and b. \((g \circ f)(x)\) Simplify the results. Find the domain of each of the results. $$ f(x)=x^{2}+7, g(x)=x^{2}-3 $$

Step-by-Step Solution

Answer

\((f\circ g)(x) = x^4 - 6x^2 + 16\), \((g\circ f)(x) = x^4 + 14x^2 + 46\). Domains: all real numbers.

Step 1: Understanding Composition

To find \((f \circ g)(x)\), we need to substitute \(g(x)\) into \(f(x)\). This means we'll replace \(x\) in \(f(x)\) with \(g(x)\). Similarly, for \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\).

Step 2: Calculate (f ◦ g)(x)

Given \(g(x) = x^2 - 3\), substitute \(g(x)\) into \(f(x)\): \[(f \circ g)(x) = f(g(x)) = f(x^2 - 3) = (x^2 - 3)^2 + 7\] Simplify the expression: \[(x^2 - 3)^2 = x^4 - 6x^2 + 9\] Thus, \[(f \circ g)(x) = x^4 - 6x^2 + 9 + 7 = x^4 - 6x^2 + 16\]

Step 3: Calculate (g ◦ f)(x)

Given \(f(x) = x^2 + 7\), substitute \(f(x)\) into \(g(x)\): \[(g \circ f)(x) = g(f(x)) = g(x^2 + 7) = (x^2 + 7)^2 - 3\] Simplify the expression: \[(x^2 + 7)^2 = x^4 + 14x^2 + 49\] Thus, \[(g \circ f)(x) = x^4 + 14x^2 + 49 - 3 = x^4 + 14x^2 + 46\]

Step 4: Determine the Domain

The domain of a polynomial function is all real numbers since there are no restrictions such as square roots or denominators. Thus, the domain for both \((f \circ g)(x) = x^4 - 6x^2 + 16\) and \((g \circ f)(x) = x^4 + 14x^2 + 46\) is all real numbers.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving only non-negative integer powers of the variable. In simpler terms, they consist of terms like \(x^2\), \(x^3\), and so on, combined using operations such as addition, subtraction, and multiplication. These functions do not involve roots, exponents, or division by variables.

Key characteristics of polynomial functions include:

• They can have one or more terms, such as \(x^4 + 3x^2 - 2\) (a polynomial of degree 4).
• The highest power of the variable gives the degree of the polynomial, which influences the function's behavior and graph.
• Coefficients are constants multiplied by each term's variable's power.

In our exercise, each function \(f(x) = x^2 + 7\) and \(g(x) = x^2 - 3\) is a polynomial. Substituting one polynomial into another results in complex polynomials like \((f \circ g)(x)\) and \((g \circ f)(x)\).

Composing functions like this helps explore how varying formulations influence the overall polynomial structure.

Domain of Functions

The domain of a function represents all the possible values of \(x\) that the function can accept. For polynomial functions, the situation is quite straightforward—since there are no denominators, square roots, or other expressions that impose restrictions, the domain usually includes all real numbers.

A polynomial function's unrestricted domain allows for ease of calculation and simplicity in understanding its behavior across all values. In our example:

• The functions \((f \circ g)(x) = x^4 - 6x^2 + 16\) and \((g \circ f)(x) = x^4 + 14x^2 + 46\) are both polynomials.
• This means their domain is \((-\infty, \infty)\), or all real numbers, because no inputs cause the function to be undefined.

Having an understanding of the domain is essential because it helps in determining the potential inputs for the function without encountering undefined behavior.

Simplifying Expressions

Simplifying expressions is a key algebraic skill where we aim to rewrite a complex expression in a simpler or more manageable form. This involves performing operations such as expanding polynomials, combining like terms, and using basic arithmetic to consolidate the expression into the simplest possible format.

For example, when simplifying the composition \((f \circ g)(x) = (x^2 - 3)^2 + 7\), the following steps occur:

• First, expand \((x^2 - 3)^2\) to get \(x^4 - 6x^2 + 9\).
• Then add 7, resulting in \(x^4 - 6x^2 + 16\).

The same principles apply when simplifying \((g \circ f)(x)\) by expanding \((x^2 + 7)^2\) to \(x^4 + 14x^2 + 49\), then subtracting 3 to result in \(x^4 + 14x^2 + 46\).

Simplification transforms a potentially complicated expression into one that is easier to interpret, operate with, and communicate.