Problem 106 - Applied Calculus

Question

a. Is the composition of two quadratic functions always a quadratic function? [Hint: Find the composition of \(f(x)=x^{2}\) and \(g(x)=x^{2}\).] b. Is the composition of two polynomials always a polynomial?

Step-by-Step Solution

Answer

a. No, the composition of two quadratics is not always quadratic. b. Yes, the composition of two polynomials is always a polynomial.

Step 1: Define the Composite Function

Given the functions \( f(x) = x^2 \) and \( g(x) = x^2 \). For part (a), we need to find the composition \( (f \circ g)(x) \) which means we will substitute \( g(x) \) into \( f(x) \).

Step 2: Substitute the Inner Function

Substitute \( g(x) = x^2 \) into \( f(x) = x^2 \). This gives \( f(g(x)) = f(x^2) = (x^2)^2 \).

Step 3: Simplify the Expression

Simplify \( (x^2)^2 \) to get \( x^4 \). Thus, the composition \( f(g(x)) = x^4 \).

Step 4: Conclude for Quadratic Functions

A quadratic function is of the form \( ax^2 + bx + c \). The resulting function \( x^4 \) is a polynomial of degree 4, not a quadratic. Thus, the composition of two quadratic functions is not always quadratic.

Step 5: Define Polynomial Composition

For part (b), consider two polynomial functions \( p(x) \) and \( q(x) \). Their composition \( (p \circ q)(x) = p(q(x)) \) involves replacing \( x \) in \( p(x) \) with the polynomial \( q(x) \).

Step 6: Composition of Polynomials

Substituting any polynomial into another always results in a polynomial since the operations involved - addition and multiplication - maintain the polynomial nature.

Step 7: Conclude for Polynomial Functions

Since the operations of addition and multiplication in the composition do not disrupt the polynomial form, the composition of two polynomials always results in another polynomial.

Key Concepts

Quadratic Functions

Quadratic functions are expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These functions describe parabolas when graphed, which can open upwards or downwards depending on the sign of \(a\).
Quadratic functions have several important features like the vertex (the highest or lowest point of the parabola), the axis of symmetry (a vertical line that divides the parabola into two mirror-image halves), and the roots or zeros (the x-values where the parabola crosses the x-axis).
Understanding these characteristics helps in solving equations and analyzing function behavior. However, when it comes to function composition, things become quite different, as seen with quadratic functions.

Composite Function

A composite function involves taking two functions and combining them in a way that the output of one function becomes the input of another function. If you have two functions, say \(f(x)\) and \(g(x)\), their composition is denoted as \((f \circ g)(x)\) or \(f(g(x))\).
To form a composite function, you perform the inner function \(g(x)\) first, then use its result as the input for the outer function \(f(x)\).
As in the exercise, if both functions are quadratic, the composite function may not be quadratic. For example, \(f(x) = x^2\) and \(g(x) = x^2\) result in \((f \circ g)(x) = x^4\), turning into a quartic, not quadratic.

Degree of Polynomials

The degree of a polynomial is the highest power of the variable in the expression. For example, \(4x^3 + 3x^2 - 2x + 7\) is a polynomial of degree 3 because the term \(4x^3\) has the highest exponent. The degree gives an indication of the polynomial's complexity.
In a composite function formed by polynomials, the degree is usually the product of the degrees of the individual functions. If you compose two quadratic functions with degree 2 each, the resulting degree is \(2 \times 2 = 4\).
This explains why composing two quadratic functions can create a quartic polynomial, as the degree increases and affects the function's shape and complexity.

Function Operations

Function operations include addition, subtraction, multiplication, and composition of functions. These operations enable the creation of new functions from existing ones, extending the ways functions can be used and interpreted.
Composite functions are particularly interesting as they involve plugging one function into another, a more advanced operation than the typical arithmetic functions. However, like other operations, they maintain the nature of the functions; thus, when you compose polynomials through composition, you always get another polynomial.
This is because polynomial functions are closed under composition, meaning that combining them does not change the basic functional form. This makes polynomial compositions reliable for forming new polynomial functions.