Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find \(f(g(h(x)))\) and \(g(h(f(x)))\). $$ f(x)=3 x^{2}+x, g(x)=x+1, h(x)=\frac{2}{3 x} $$

Step-by-Step Solution

Answer

The composite functions are \(f(g(h(x))) = \frac{4}{3x^2}+\frac{2}{3x}+6\) and \(g(h(f(x))) = \frac{2}{9x^2+3x} + 1\)

Step 1: Find \(f(g(h(x)))\)

Start by substituting \(h(x)\) into \(g(x)\) as it is the innermost function. After that, you'll substitute the result of \(g(h(x))\) into \(f(x)\). So first, let's find \(g(h(x))= g\left(\frac{2}{3 x}\right) = \frac{2}{3 x} + 1\). Now, substitute this into \(f(x)\) so \(f\left( g\left( h(x) \right) \right) = f\left(\frac{2}{3 x} + 1\right) = 3\left(\frac{2}{3 x} + 1\right)^2+\frac{2}{3 x} + 1 = 3\left(\frac{4}{9x^2}+\frac{2}{3x} + 1\right)+\frac{2}{3x}+1 = \frac{4}{3x^2}+2+3+\frac{2}{3x}+1 = \frac{4}{3x^2}+\frac{2}{3x}+6\).

Step 2: Find \(g(h(f(x)))\)

Similarly, begin by substituting \(f(x)\) into \(h(x)\) as it is the innermost function. Then, you'll substitute the result of \(h(f(x))\) into \(g(x)\). So first, let's find \(h(f(x)) = h\left(3 x^2 +x\right) = \frac{2}{3 \left(3 x^2 +x\right)} = \frac{2}{9x^2+3x}\). Now, substitute this into \(g(x)\) so \(g\left( h\left( f(x) \right) \right) = g\left(\frac{2}{9x^2+3x}\right) = \frac{2}{9x^2+3x} + 1\

Key Concepts

Function Operations

Function operations are akin to arithmetic operations, but they are performed with functions instead of numbers. A function, denoted as a letter like 'f', 'g', or 'h', represents a relationship between a set of inputs and a set of possible outputs. Just as you can add, subtract, multiply, or divide numbers, functions can be added, subtracted, multiplied, divided, or composed.
Function composition is a key operation where you apply one function to the results of another function. Understanding how to combine functions using these operations is essential in mathematics. Visualizing this concept helps: Imagine a machine 'f' that transforms a ball 'x' into a new shape. Following that, another machine 'g' modifies this new shape even further. The combination of both machines' operations on 'x' is what we call the composition 'g(f(x))'.

Substituting Functions

Substituting functions, or function substitution, is the process where you replace the input of a function with another function. This is like changing the recipe for a dish by using a different ingredient that has been altered by another recipe. For example, if you have functions 'f' and 'g', and you are tasked to find 'f(g(x))', you would first calculate the result of 'g(x)' and then use that result as the input for function 'f'.
It’s crucial to work from the inside out, dealing first with the innermost function. This hierarchical approach may remind one of nesting dolls, where you open the smallest doll first before you can get to the next size.

Calculating Function Composition

Calculating function composition involves evaluating the inner function and using its output as the input for the outer function. Follow the order precisely to avoid any mistakes. You typically start with the innermost function, which in our exercise was 'h', and substitute its output into the next function 'g'. Then, take the output from 'g(h(x))' and use it as the input for 'f'. This chained process requires careful substitution and simplification after each step.
Example:
Given the composition 'f(g(h(x)))', first calculate 'h(x)', then substitute that into 'g(x)' to find 'g(h(x))', and, finally, put the result into 'f(x)' to get 'f(g(h(x)))'. This can be imagined as a multi-layered process where each function acts as a filter, transforming the variable 'x' step by step until the composition is fully calculated.

Algebraic Functions

Algebraic functions involve operations that include polynomial, rational, and root functions. They typically consist of a finite combination of algebraic operations - addition, subtraction, multiplication, division, and taking roots - on variables. These functions are fundamental in algebra and are the building blocks for more complex mathematical concepts.
In the given exercise, functions 'f', 'g', and 'h' are all algebraic, with 'f' being a polynomial function, 'g' a linear function, and 'h' a rational function. Students should be familiar with operations on these types of functions to understand how to solve the function compositions correctly. Mastery over algebraic functions and their compositions can greatly enhance one's problem-solving toolkit in mathematics.