Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find \(f(g(h(x)))\) and \(g(h(f(x)))\). $$ f(x)=\frac{1}{x}, g(x)=\sqrt{x}, h(x)=x-3 $$

Step-by-Step Solution

Answer

The composition of \(f(g(h(x)))\) results to \(f(g(h(x))) = \frac{1}{\sqrt{x-3}}\) and composition of \(g(h(f(x)))\) results to \(g(h(f(x))) = \sqrt{\frac{1}{x}-3}\)

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

Firstly, calculate \(h(x)\), substituting \(x\) with the given value. Once \(h(x) = x - 3\) is calculated, substitute this value into the next function in the expression, \(g(x) = \sqrt{x}\), to get \(g(h(x)) = \sqrt{x-3}\). Finally, substitute this result into \(f(x)\), resulting in \(f(g(h(x))) = \frac{1}{\sqrt{x-3}}\).

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

Start with calculating \(f(x)\), which as per the given function, is \(f(x) = \frac{1}{x}\). Now substitute this result into \(h(x)\) to find \(h(f(x)) = \frac{1}{x}-3\). Finally, carry the results over to \(g(x) = \sqrt{x}\), which after substitution will result in \(g(h(f(x))) = \sqrt{\frac{1}{x}-3}\).

Key Concepts

Composition of Functions

Understanding the composition of functions is essential in calculus, as it involves combining two or more functions to form a new function. In the context of our exercise, the composition of functions can be seen when we are asked to find expressions like f(g(h(x))) and g(h(f(x))). Composition often requires careful substitution where one function's output becomes the input of another.

For instance, to find f(g(h(x))), we started by figuring out h(x) = x - 3. The result of h(x) is then substituted into g(x), resulting in g(h(x)) = \( g(h(x)) = \sqrt{x-3} \). Lastly, this outcome serves as the input for f(x), leading us to f(g(h(x))) = \( \frac{1}{\sqrt{x-3}} \). To understand this better, picture functions as machines, where you insert something into one machine and use its output for the next, creating a chain of operations.

Function Operations

Function operations include various ways in which functions can be combined or manipulated, such as addition, subtraction, multiplication, division, and composition. Our focus, in this case, is the operation of function composition, but understanding all operation types is crucial.

When dealing with function operations, it's important to perform each operation in the correct order, similar to the order of operations in arithmetic (PEMDAS/BODMAS). In the exercise, the composition operation tells us to work from the innermost function outward, ensuring that each function is fully evaluated before its output is used in the next function.

Inverse Functions

While the concept of inverse functions isn't directly applied in this exercise, grasping it can deepen our understanding of functions as a whole. An inverse function essentially undoes what the original function does. For example, if we have a function that adds 3 to any number, the inverse would subtract 3, returning us to the original number.

If f(x) is a function, then its inverse, denoted as f^{-1}(x), will satisfy the condition that f(f^{-1}(x)) = x and f^{-1}(f(x)) = x for all x in the domain of f. An easy way to verify if two functions are inverses is to compose them and see if the result simplifies to x. It's also important to note that not all functions have inverses, particularly those that are not one-to-one (each x maps to a unique y and vice versa).