Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Decompose the functions by finding functions \(f(x)\) and \(g(x)\), \(f(x) \neq x\) and \(g(x) \neq x\), such that \(h(x)=f(g(x))\). $$ h(x)=\sqrt{x^{2}+1} $$

Step-by-Step Solution

Answer

The decomposition of the function can be represented as \(h(x) = f(g(x))\) where \(f(x) = \sqrt{x}\) and \(g(x) = x^{2} + 1\).

Step 1: Initialize

The given function is \(h(x) = \sqrt{x^{2} + 1}\). We have to express it in the form of \(f(g(x))\), with \(f(x) \neq x\) and \(g(x) \neq x\).

Step 2: Determine g(x)

We first identify the inner function in \(h(x)\). The expression inside the square root function, \(x^{2} + 1\), seems to be the 'inner' function because it is the input to another function. Therefore, we can set \(g(x) = x^{2} + 1\). It is clearly a function different than \(x\).

Step 3: Determine f(x)

The 'outer' function in \(h(x)\) appears to be the square root function that operates on the result of \(g(x)\). Therefore, we can set \(f(x) = \sqrt{x}\). This function too is different than \(x\).

Key Concepts

Composition of Functions

The composition of functions involves creating a new function by applying one function to the result of another. In mathematical terms, if you have two functions, say \( f(x) \) and \( g(x) \), their composition is represented as \( f(g(x)) \). In this setup, the output of the function \( g(x) \) becomes the input for the function \( f(x) \). It is a way to combine functions to make more complex mappings from one set to another.
For instance, in the given problem, we have a function \( h(x) = \sqrt{x^2 + 1} \). By finding functions \( f(x) \) and \( g(x) \) such that \( h(x) = f(g(x)) \), we are performing composition. \( g(x) \) is applied first, and its result becomes the input to \( f(x) \).

Function Operations

Function operations involve various ways of combining functions to produce new ones. Besides composition, there are other operations such as addition, subtraction, multiplication, and division. Each of these operations combines the outputs of functions in different ways:

• Addition: \((f + g)(x) = f(x) + g(x)\)
• Subtraction: \((f - g)(x) = f(x) - g(x)\)
• Multiplication: \((f \cdot g)(x) = f(x) \cdot g(x)\)
• Division: \((f / g)(x) = \frac{f(x)}{g(x)}\)\, where \(g(x) eq 0\)

However, it's important to note that composition is distinct because it nests functions instead of merely combining their outputs.

Inner and Outer Functions

When dealing with functional composition, a clear understanding of inner and outer functions is essential. Here's what these terms mean:

• Inner Function: This is the function that operates first or is inside another function. In \( h(x) = f(g(x)) \), \( g(x) \) is the inner function. For our example, \( g(x) = x^2 + 1 \).
• Outer Function: This is the function that operates second, applying to the result of the inner function. In \( h(x) = f(g(x)) \), \( f(x) \) is the outer function. Here, \( f(x) = \sqrt{x} \).

Recognizing the roles of inner and outer functions can deeply enhance understanding of complex mathematical expressions and their behavior.

Mathematical Expressions

Mathematical expressions consist of numbers, variables, and functions, combined to represent quantities or relationships. These expressions can be simple, like \( x + 2 \), or complex, like \( \sqrt{x^2 + 1} \). An expression like \( \sqrt{x^2 + 1} \) has distinct parts:

• The square root function is applying a mathematical operation.
• The expression \( x^2 + 1 \) serves as the operand for that operation.

Here, identifying each component helps in breaking down the expression into manageable parts. This is especially useful for evaluating expressions or solving equations, making complex relationships easier to manage.