Problem 44 - Algebra for College Students

Question

If \(f(x)=3 x^{2}-2 x-1\) and \(g(x)=x\), find \(f \circ g\) and \(g \circ f\). (Recall that we have previously named \(g(x)=x\) the "identity function.")

Step-by-Step Solution

Answer

Both \(f \circ g\) and \(g \circ f\) equal \(3x^2 - 2x - 1\).

Step 1: Understand Composition of Functions

The composition of two functions, say \(f\) and \(g\), denoted as \((f \circ g)(x)\), means applying \(g\) first and then \(f\) on the result. Conversely, \((g \circ f)(x)\) means applying \(f\) first and then \(g\) to the result.

Step 2: Compute f(g(x))

Since \(g(x) = x\) is the identity function, \((f \circ g)(x) = f(g(x)) = f(x)\). We plug in \(x\) into \(f(x)\): \(f(x) = 3x^2 - 2x - 1\). Thus, \((f \circ g)(x) = 3x^2 - 2x - 1\).

Step 3: Compute g(f(x))

For \((g \circ f)(x) = g(f(x))\), we need to substitute \(f(x)\) into \(g\), which is the identity. Thus, \(g(f(x)) = f(x) = 3x^2 - 2x - 1\).

Key Concepts

Identity Function

In mathematics, the identity function is a special kind of function that simply returns the input value without making any changes. It is represented as \( g(x) = x \). This means that for any input \( x \), the output is also \( x \). The identity function is fundamental because when it is composed with any other function, it doesn't alter the function's behavior or value.

Consider the composition \((f \circ g)(x)\), where \(f\) is any given function and \(g(x) = x\). In this case, you apply \(g\), which leaves \(x\) unchanged, before applying \(f\). This results in \((f \circ g)(x) = f(x)\). Similarly, for \((g \circ f)(x)\), you apply \(f\) first and then \(g\), resulting once again in \(f(x)\) because \(g(x)\) doesn't modify the output. Thus, the identity function behaves like a mirror, reflecting the input exactly as it is. It is essential because it serves as a neutral element in function composition.

Polynomial Function

A polynomial function is an expression involving a sum of powers of one or more variables multiplied by coefficients. The general form for a single-variable polynomial function is \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants, and \(n\) is a non-negative integer.

In our original exercise, we have the polynomial function \(f(x) = 3x^2 - 2x - 1\). Here, the highest power of \(x\) is 2, making it a quadratic polynomial. Such functions are known for their characteristic parabola shape when graphed.

• The leading coefficient is 3, which affects the width and direction of the parabola.
• Since the leading coefficient is positive, the parabola opens upwards.

Understanding polynomial functions is crucial as they form the building blocks for more complex functions and many phenomena in real-world applications like physics and engineering.

Function Operations

Function operations encompass a variety of ways to combine functions to create new ones. The most common operations are addition, subtraction, multiplication, division, and composition of functions. The composition of functions, as seen in the exercise, involves combining two functions such that the output from the first function becomes the input for the second function.

To better understand function composition, imagine it as a step-by-step process:

• Start with an input \(x\).
• Apply the first function to \(x\) to get an intermediate result.
• Use this intermediate result as the input to the second function.

This is seen in \((f \circ g)(x)\) and \((g \circ f)(x)\) from the exercise, where the identity function \(g(x) = x\) simplifies the composition, as it doesn't change the result of \(f(x)\).

Mastering function operations is pivotal for students, as this understanding provides a foundation for more advanced math topics like calculus and differential equations.