Problem 2 - Applied Calculus

Question

In Problems \(1-3,\) find the following: (a) \(\quad f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=x-2 \text { and } g(x)=x^{2}+8$$

Step-by-Step Solution

Answer

(a) \( f(g(x)) = x^2 + 6 \), (b) \( g(f(x)) = x^2 - 4x + 12 \), (c) \( f(f(x)) = x - 4 \).

Step 1: Find f(g(x))

To find \( f(g(x)) \), substitute \( g(x) \) into the function \( f(x) \).Given, \( g(x) = x^2 + 8 \). Substitute it into \( f(x) \):\[ f(g(x)) = f(x^2 + 8) \]Using the function definition \( f(x) = x - 2 \), we replace \( x \) in \( f(x) \) with \( x^2 + 8 \):\[ f(g(x)) = (x^2 + 8) - 2 \]Simplify the expression:\[ f(g(x)) = x^2 + 6 \].

Step 2: Find g(f(x))

To find \( g(f(x)) \), substitute \( f(x) \) into the function \( g(x) \).Given, \( f(x) = x - 2 \). Substitute it into \( g(x) \):\[ g(f(x)) = g(x - 2) \]Using the function definition \( g(x) = x^2 + 8 \), replace \( x \) in \( g(x) \) with \( x - 2 \):\[ g(f(x)) = (x - 2)^2 + 8 \]Simplify the expression:First expand \( (x - 2)^2 \):\[ (x - 2)^2 = x^2 - 4x + 4 \]Substitute back:\[ g(f(x)) = x^2 - 4x + 4 + 8 \]Combine like terms:\[ g(f(x)) = x^2 - 4x + 12 \].

Step 3: Find f(f(x))

To find \( f(f(x)) \), substitute \( f(x) \) into the function \( f(x) \).Given, \( f(x) = x - 2 \). Substitute it into the same function \( f(x) \):\[ f(f(x)) = f(x - 2) \]Using \( f(x) = x - 2 \), replace \( x \) in \( f(x) \) with \( x - 2 \):\[ f(f(x)) = (x - 2) - 2 \]Finally, simplify the expression:\[ f(f(x)) = x - 4 \].

Key Concepts

Function Composition

Function composition is an incredibly useful tool in mathematics, allowing us to combine multiple functions into a single new function. Function composition is denoted by combining two function names, such as \( f(g(x)) \) or \( g(f(x)) \).
To compose two functions, substitute one function into another.

• For example, if we have functions \( f(x) \) and \( g(x) \), the expression \( f(g(x)) \) means that we first apply \( g \) to \( x \), and then apply \( f \) to the result.
• Similarly, \( g(f(x)) \) indicates that we first apply \( f \) to \( x \), then apply \( g \) to the new result.

This process of substitution allows us to explore how different functions can interact with each other. It also helps us analyze complex systems where multiple transformations are involved.
To enhance your understanding of function composition:

• Practice by taking more examples and finding the composition of two or more functions.
• Focus on the order of operations, as reversing the functions can yield different results.
• Understand that each step in the composition leads to a new function which might have its own unique properties.

Algebraic Manipulation

Algebraic manipulation is the backbone of simplifying and solving composite functions. Understanding how to correctly manipulate expressions allows us to correctly break down complex problems into simpler steps.
In our solutions for the composite functions, we made use of algebraic manipulation:

• When finding \( f(g(x)) \), the expression \( x^2 + 8 - 2 \) was simplified to \( x^2 + 6 \), by combining like terms.
• For \( g(f(x)) \), we expanded the expression \((x-2)^2\) into \(x^2 - 4x + 4\) and then simplified further to \( x^2 - 4x + 12 \) by combining similar terms.

Techniques such as distributing terms, factoring, expanding expressions, and combining like terms are essential. These operations ensure that the mathematical expressions are in their simplest, most understandable form.
With regular practice, you'll become adept at recognizing which techniques to apply in different situations, making the process of solving composite functions both efficient and straightforward.

Polynomial Functions

Polynomial functions are a key concept in algebra and are frequently encountered in problems involving composite functions. In the given exercise, the function \( g(x) = x^2 + 8 \) is a polynomial function, specifically a quadratic polynomial.
Polynomials are expressions made up of variables and coefficients, connected using addition, subtraction, multiplication, and non-negative integer exponents.

• In our context, the function \( f(g(x)) = x^2 + 6 \) represents a polynomial function after the function composition \( f(g(x)) \).
• The composite \( g(f(x)) = x^2 - 4x + 12 \) is also a polynomial.

Polynomial functions have specific characteristics depending on their degree:

• A linear polynomial, like \( f(x) = x - 2 \), has a degree of 1.
• A quadratic polynomial, like \( x^2 + 8 \), has a degree of 2.

Understanding these functions and how they behave is important:

• It aids in predicting the graph of the function.
• Recognizing patterns in polynomial functions can help simplify mid to high-level algebraic problems.

When working with polynomial functions, practice identifying their degree, leading coefficient, and roots. These elements are key in understanding the broader behavior of the function.