Problem 100 - Precalculus : Building Concepts and Connections

Question

In Exercises \(97-100,\) let \(f(t)=-t^{2}\) and \(g(x)=x^{2}-1\). Find an expression for \((g \circ g)(x),\) and give the domain of \(g \circ g\).

Step-by-Step Solution

Answer

The expression for \( (g \circ g)(x) \) is \( x^{4} - 2x^{2} \) and the domain of \( g \circ g \) is all real numbers or in interval notation \( (-\infty, +\infty) \).

Step 1: Find \( (g \circ g)(x) \)

The composition of \( g \circ g \) represents the operation of first applying the function g and then applying the function g again. In other words, we plug the output of the first application of g into the second one. Thus, \( (g \circ g)(x) = g(g(x)) \). We substitute \( g(x) \) into itself. Recall that \( g(x) = x^{2} -1 \). This gives us: \( (g \circ g)(x) = g(x^{2} -1) = (x^{2} -1)^{2} -1 \). So, \( (g \circ g)(x) = x^{4} - 2x^{2} \).

Step 2: Find the domain of \( (g \circ g)(x) \)

The domain of a function is the set of all values for which the function is defined. Since \( (g \circ g)(x) = x^{4} - 2x^{2} \), a polynomial function, there are no fractions with x in the denominator and no square roots of a negative number. Therefore, the domain of \( g \circ g \) is all real numbers. In interval notation, the domain is \( (-\infty, +\infty) \).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (commonly represented as 'x') for which the function provides a defined output. Think of it as all the x-values you can plug into the function that yield a meaningful result without breaking any mathematical rules. In the exercise, the function \(g \circ g\) is a polynomial, meaning its domain is simpler to determine. For polynomial functions like \(g \circ g(x) = x^4 - 2x^2\), the domain is unrestricted because there's no division by zero, no square roots which could be negative, and no other operations that need extra caution. Essentially, you can plug any real number into \(x\). Hence, its domain is all real numbers, expressed in interval notation as \((-\infty, +\infty)\). This is a typical characteristic of polynomial functions, making them straightforward when it comes to domain analysis.

Polynomial Functions

Polynomial functions are mathematical expressions made up of terms with coefficients and variables raised to non-negative integer powers. For example, a polynomial might look like \(x^4 - 2x^2\), as seen in our exercise with \(g \circ g(x)\). Here are key features of polynomial functions:

• They're built from terms like \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where each \(a_i\) is a constant, and \(n\) is a non-negative integer.
• The highest power of x defines the polynomial's "degree." In this exercise, the degree is 4 due to the \(x^4\) term.
• Polynomials are continuous and smooth, meaning they're defined for all x-values, which directly relates to their inclusive domain.

Understanding these basic traits helps in solving and visualizing polynomial functions effectively. They often default to having an all-encompassing domain unless specifically constrained, making them versatile in problem-solving scenarios.

Operations on Functions

Operations on functions involve combining them in various ways, including addition, subtraction, multiplication, division, and composition. Function composition is crucial when working with functions, and it involves taking one function's output and using it as the input for another function.
In the exercise, the function \(g \circ g\) is found by composing \(g\) with itself. Here's a breakdown:

• Start with the function \(g(x) = x^2 - 1\).
• Substitute \(g(x)\) into itself, resulting in \(g(g(x))\).
• This gives \(g(x^2 - 1) = (x^2 - 1)^2 - 1\).
• Simplifying yields \(x^4 - 2x^2\), which is the expression for \(g \circ g(x)\).

Composition enables transforming functions into new forms, thus expanding their utility. By mastering operations like composition, you deepen your ability to manipulate and utilize different functions in mathematical problems.