Problem 84 - Elementary and Intermediate Algebra

Question

If \(f(x)=x^{2}+1\) and \(g(x)=3 x^{2}-2,\) show that \((f \circ g)(x) \neq(g \circ f)(x)\)

Step-by-Step Solution

Answer

The compositions \\( (f \circ g)(x) \\\) and \\( (g \circ f)(x) \\\) are different: \\( 9x^4 - 12x^2 + 5 \\\) vs \\( 3x^4 + 6x^2 + 1 \\\).

Step 1: Understand the Composite Functions

First, let's clarify the notation and meaning of \( (f \circ g)(x) \) and \( (g \circ f)(x) \). The notation \( (f \circ g)(x) \) means applying \( g(x) \) first, and then applying \( f(x) \) to the result. Similarly, \( (g \circ f)(x) \) means applying \( f(x) \) first, then \( g(x) \) to the result. Our task is to evaluate both expressions and compare them.

Step 2: Calculate \\( (f \circ g)(x) \\\)

Using the definition of composition, \( (f \circ g)(x) = f(g(x)) \).1. Substitute \( g(x) \) into \( f(x) \): \( g(x) = 3x^2 - 2 \), so:2. \( f(3x^2 - 2) = (3x^2 - 2)^2 + 1 \).3. Expand \( (3x^2 - 2)^2 \): \[ (3x^2 - 2)^2 = (3x^2)^2 - 2\cdot3x^2\cdot2 + 2^2 = 9x^4 - 12x^2 + 4 \].4. Add 1 to the result: \( 9x^4 - 12x^2 + 4 + 1 = 9x^4 - 12x^2 + 5 \).

Step 3: Calculate \\( (g \circ f)(x) \\\)

Now apply the function composition in the reverse order: \( (g \circ f)(x) = g(f(x)) \).1. Substitute \( f(x) \) into \( g(x) \): \( f(x) = x^2 + 1 \), so:2. \( g(x^2 + 1) = 3(x^2 + 1)^2 - 2 \).3. Expand \( (x^2 + 1)^2 \): \[ (x^2 + 1)^2 = (x^2)^2 + 2\cdot x^2\cdot1 + 1^2 = x^4 + 2x^2 + 1 \].4. Multiply the result by 3: \[ 3(x^4 + 2x^2 + 1) = 3x^4 + 6x^2 + 3 \].5. Subtract 2: \( 3x^4 + 6x^2 + 3 - 2 = 3x^4 + 6x^2 + 1 \).

Step 4: Compare the Results

We have determined the expressions for each composition:1. \( (f \circ g)(x) = 9x^4 - 12x^2 + 5 \)2. \( (g \circ f)(x) = 3x^4 + 6x^2 + 1 \)Clearly, these two expressions are different because the coefficients of \( x^4 \) and \( x^2 \) terms, as well as the constant terms, do not match.

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics, often represented by the symbol \( \circ \). It involves applying one function to the result of another function. This is akin to doing one operation and then using that result to perform another operation immediately afterward.

• For \( (f \circ g)(x) \), you first apply \( g(x) \), and then apply \( f \) to the output of \( g \).
• For \( (g \circ f)(x) \), you begin with \( f(x) \) and apply \( g \) to the result of \( f \).

The order of applying functions is crucial because it can drastically alter the outcome. Understanding this order helps to highlight why \( (f \circ g)(x) \) might not equal \( (g \circ f)(x) \). In the example from the exercise, you can see how a change in order changes the entire polynomial expression created by these compositions.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers and combined through addition, subtraction, and multiplication. They are some of the most straightforward and widely-used mathematical tools.
Let’s look at some key characteristics:

• Polynomial functions, like \( f(x)=x^2+1 \) and \( g(x)=3x^2-2 \), consist of terms. Each term is the product of a constant and a variable raised to a power.
• Terms in a polynomial expression are ordered by descending power of the variable, and the coefficients are the numbers in front of these terms.

Understanding how to manipulate polynomials, such as expanding \((x^2 + 1)^2\) or \((3x^2 - 2)^2\), is essential in evaluating compositions. When functions are composed, each polynomial essentially transforms, combining into a new polynomial structure, highlighting the nuances of function composition.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (addition, subtraction, multiplication, and sometimes division). Function composition deals extensively with these expressions as you substitute one expression into another.
Here are some ways algebraic expressions play a crucial role:

• When you compose functions like in the exercise, you substitute one expression, like \( g(x)=3x^2-2 \), into another expression for \( f(x) \).
• This substitution creates layered algebraic expressions that need simplifying and rearranging by expanding powers and combining like terms, as shown in the step-by-step solution.
• Working with algebraic expressions requires attention to detail, especially with distribution and exponent rules, ensuring each component is correctly applied.

Breaking down each step of an algebraic expression's manipulation helps in demystifying complex results, as when you see how different compositions of functions like \( (f \circ g)(x) \) and \( (g \circ f)(x) \) yield unique and different outcomes.