Problem 50 - Brief Applied Calculus

Question

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=x^{8} ; \quad g(x)=2 x+5 $$

Step-by-Step Solution

Answer

a. \((2x + 5)^8\), b. \(2x^8 + 5\), c. \(x^{64}\)

Step 1: Substitute g(x) into f(x) for f(g(x))

First, we need to find the composition of functions for \(f(g(x))\). To do this, substitute \(g(x) = 2x + 5\) into \(f(x) = x^8\). This gives us \(f(g(x)) = (2x + 5)^8\).

Step 2: Simplify g(f(x)) expression

Next, find \(g(f(x))\) by substituting \(f(x) = x^8\) into \(g(x) = 2x + 5\). This results in \(g(f(x)) = 2(x^8) + 5\), which simplifies to \(2x^8 + 5\).

Step 3: Substitute f(x) into itself for f(f(x))

To find \(f(f(x))\), substitute \(f(x) = x^8\) into itself. This means \(f(f(x)) = (x^8)^8\), which simplifies to \(x^{64}\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the process of reorganizing and rewriting expressions to achieve a desired form. It's essential in simplifying composite functions, as seen in the exercise. Consider the expression for the function composition \( f(g(x)) \). Here, \( f(x) = x^8 \) and \( g(x) = 2x + 5 \). To find \( f(g(x)) \), plug \( g(x) \) into \( f(x) \), resulting in \((2x + 5)^8\). This power can be expanded using algebraic rules, although in many cases, we leave it in this form to avoid excessive complexity.

• Substitution: You replace the variable in one function with another expression to achieve the composition.
• Simplification: Once you've substituted, use algebraic laws to simplify if possible.

Understanding how to properly substitute and manipulate algebraic expressions is crucial for solving composite functions and finding simpler forms.

Composite Functions

Composite functions involve combining two or more functions to create a new function. This is evident with the functions \( f(x) = x^8 \) and \( g(x) = 2x + 5 \) in the exercise. By creating composite functions like \( f(g(x)) \), \( g(f(x)) \), and \( f(f(x)) \), you evaluate a function's output as another function's input.

• Order Matters: The sequence of composition is critical. For instance, \( f(g(x)) \) and \( g(f(x)) \) generally produce different results.
• Nested Application: You apply each function to the next, working from the inside out.

To deepen your comprehension, remember that the composition reflects real-world processes where input changes impact subsequent systems or calculations.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations. They represent values or computations, such as the expressions seen in our functions. For example, \((2x + 5)^8\) and \(2x^8 + 5\) are mathematical expressions formed by substituting one function into another.

• Variables and Constants: These components allow flexibility and fixed values within expressions.
• Operations: Addition, multiplication, and exponentiation build the structure of expressions.

Expressions form the backbone of algebra and calculus, enabling the representation of varied quantities and relationships through symbols and numbers.