Problem 22 - The Calculus with Analytic Geometry

Question

Functions \(f\) and \(g\) are defined. In each exercise define \(f \circ g\), and determine all values of \(x\) for which \(f \circ g\) is continuous. $$ f(x)=x^{2} ; g(x)=x^{2}-3 $$

Step-by-Step Solution

Answer

The function \(f \circ g(x) = x^{4} - 6x^{2} + 9\) is continuous for all \(x\).

Step 1: Understand the Composition of Functions

To find the composition of the functions, we need to understand how to combine the functions. The composition of two functions, denoted as \(f \circ g\), means \(f(g(x))\).

Step 2: Substitute g(x) into f(x)

Given \(f(x) = x^{2}\) and \(g(x) = x^{2} - 3\), we substitute \(g(x)\) into \(f(x)\). Thus, \(f(g(x)) = f(x^{2} - 3)\).

Step 3: Evaluate the Composition

Substitute \(x^{2} - 3\) into \(f(x)\). This gives us \(f(x^{2} - 3) = (x^{2} - 3)^{2}\).

Step 4: Simplify the Expression

Simplify the expression \(f(x^{2} - 3)\). We get: \((x^{2} - 3)^{2} = x^{4} - 6x^{2} + 9\). Thus, \(f \circ g(x) = x^{4} - 6x^{2} + 9\).

Step 5: Determine Continuity

Both \(f(x) = x^{2}\) and \(g(x) = x^{2} - 3\) are polynomials, which are continuous for all real numbers. Therefore, \(f \circ g(x) = x^{4} - 6x^{2} + 9\) is also a polynomial and is continuous for all real numbers.

Key Concepts

continuity

In mathematics, continuity is a fundamental concept that ensures a function behaves predictably and smoothly without sudden breaks or jumps. To determine if a function is continuous, we need to verify that it meets three crucial conditions: the function must be defined at a point, the limit must exist at that point, and the function's value at that point must equal its limit. Essentially, a function is continuous if you can draw it without lifting your pen from the paper.
For polynomial functions, continuity is quite straightforward. Polynomials are continuous everywhere. This means that they don't have any gaps, jumps, or holes in their graphs. Since both functions in our exercise,  \(f(x) = x^{2}\) and \(g(x) = x^{2} - 3\), are polynomials, they are continuous for all real numbers.
Composite functions inherit the properties of the functions they're made from. Since both \(f(x)\) and \(g(x)\) are continuous everywhere, their composition \(f \bigcirc g(x) = x^{4} - 6x^{2} + 9\) will also be continuous for all real numbers. This means the function behaves predictably at every point on the real number line.

polynomial functions

Polynomial functions are expressions involving variables raised to whole number exponents and having coefficients. Examples include \(f(x) = x^{2}\), \(g(x) = x^{2} - 3\), and other forms like \(x^{3} - 4x + 5\). Their general form is \(P(x) = a_n x^n + a_{n-1} x^{n-1} +...+ a_1 x + a_0\), where \(a_n, a_{n-1},..., a_0\) are constants.
One key characteristic of polynomial functions is their smooth and unbroken graphs, which make them continuous over all real numbers. They do not possess any sudden jumps, gaps, or vertical asymptotes. Their behavior is easy to predict and analyze.
Functions like \(f(x) = x^{2}\) and \(g(x) = x^{2} - 3\) are examples of quadratic polynomials. Quadratic polynomials generally form parabolic shapes when graphed. When we evaluate \(g(x) = x^{2} - 3\) and then substitute it into another polynomial like \(f(x) = x^{2}\), we still obtain a polynomial, as seen in our exercise.

composite functions

Composition of functions refers to applying one function to the results of another. It's a way of combining two functions such that the output of the first function becomes the input of the second. This is denoted as \(f \bigcirc g(x)\), which means \(f(g(x))\).
When working with composite functions, it's important to understand each individual function first and how they interact with each other. Let's break it down based on the exercise:

• Given \(f(x) = x^{2}\) and \(g(x) = x^{2} - 3\).
• To find \(f \bigcirc g(x)\), we substitute \(g(x)\) into \(f(x)\).
• This gives us \(f(g(x)) = f(x^{2} - 3)\).
• Substitute \(x^{2} - 3\) into \(f(x)\) to get \((x^{2} - 3)^{2}\).

The simplified expression becomes \(x^{4} - 6x^{2} + 9\), which is a new polynomial formed by the composite function \(f \bigcirc g(x)\). Since both \(f(x)\) and \(g(x)\) are continuous and polynomials, their composite function is also continuous over all real numbers.
This process demonstrates how we combine functions and verify their properties like continuity by analyzing their composition.