Problem 49 - 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. \(f(f(x))\) $$ f(x)=x^{5} ; g(x)=7 x-1 $$

Step-by-Step Solution

Answer

a. \( f(g(x)) = (7x - 1)^5 \), b. \( g(f(x)) = 7x^5 - 1 \), c. \( f(f(x)) = x^{25} \).

Step 1: Determine Expression for \( f(g(x)) \)

To find \( f(g(x)) \), we need to substitute \( g(x) \) into \( f(x) \). The function \( f(x) = x^5 \) means we replace \( x \) with \( g(x) \) in this function. Since \( g(x) = 7x - 1 \), we have:\[ f(g(x)) = (7x - 1)^5 \].

Step 2: Determine Expression for \( g(f(x)) \)

To find \( g(f(x)) \), we will substitute \( f(x) \) into \( g(x) \). The function \( g(x) = 7x - 1 \) requires replacing \( x \) with \( f(x) \). Knowing that \( f(x) = x^5 \), we have:\[ g(f(x)) = 7(x^5) - 1 = 7x^5 - 1 \].

Step 3: Determine Expression for \( f(f(x)) \)

For \( f(f(x)) \), substitute \( f(x) \) into itself. Since \( f(x) = x^5 \), it means replacing \( x \) with \( f(x) \) in \( f(x) \): \[ f(f(x)) = (x^5)^5 = x^{25} \].

Key Concepts

Composite Functions

Composite functions are created when one function is applied to the result of another function. This is like a chain reaction; you start with one function and then plug its output into another function to get a new result.

For example, in the exercise provided, the functions are given as \( f(x) = x^5 \) and \( g(x) = 7x - 1 \). If you want to find \( f(g(x)) \), you substitute \( g(x) \) into \( f(x) \). This results in the expression \( (7x - 1)^5 \).

Similarly, for \( g(f(x)) \), substitute the function \( f(x) \) into \( g(x) \). This gives \( 7x^5 - 1 \). Composite functions allow you to combine functions for more complex operations, enabling a wide range of applications in both simple and advanced mathematics.

Algebraic Functions

Algebraic functions involve operations such as addition, subtraction, multiplication, division, and exponentiation on variables and constants. These are fundamental to solving equations and performing calculations in algebra.

In the exercise example, both \( f(x) = x^5 \) and \( g(x) = 7x - 1 \) are algebraic functions. \( f(x) \) is a power function, where an exponent is applied to the variable. \( g(x) \) is a linear function, showing a straightforward computation using multiplication and subtraction.

Understanding algebraic functions is crucial for identifying how different mathematical operations can be applied to variables in order to solve equations and other mathematical problems. They form the backbone for many areas of higher mathematics, from calculus to linear algebra.

Function Notation

Function notation is a helpful way to express algebraic functions clearly. It allows mathematicians and students to convey the output of a function for a given input without ambiguity.

Notation such as \( f(x) \) can be understood as "the value of function \( f \) at \( x \)". It's a way of labeling functions that makes it easier to reference them, especially when working with multiple functions, as in our exercise.

In our example, \( f(g(x)) \) indicates that \( g(x) \) is used as the input to \( f \). Similarly, \( g(f(x)) \) does the reverse process, showing a sequence in which functions are to be executed. Understanding and using function notation effectively is essential for working with complex mathematical operations and demonstrates a key part of mathematical communication.