Problem 51 - Applied Calculus

Question

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=\frac{1}{x} ; \quad g(x)=x^{2}+1 $$

Step-by-Step Solution

Answer

a. \( f(g(x)) = \frac{1}{x^2 + 1} \); b. \( g(f(x)) = \frac{1}{x^2} + 1 \); c. \( f(f(x)) = x \).

Step 1: Understand the Functions

We are given two functions: \( f(x) = \frac{1}{x} \) and \( g(x) = x^2 + 1 \). Our task is to find the compositions of these functions in three different ways.

Step 2: Compute f(g(x))

To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \). So, \( f(g(x)) = f(x^2 + 1) = \frac{1}{x^2 + 1} \). This is because wherever there is an \( x \) in \( f(x) \), we now put \( g(x) \).

Step 3: Compute g(f(x))

To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \). Thus, \( g(f(x)) = g\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^2 + 1 = \frac{1}{x^2} + 1 \). This reflects substituting \( f(x) \) into every occurrence of \( x \) in \( g(x) \).

Step 4: Compute f(f(x))

To find \( f(f(x)) \), we substitute \( f(x) \) into itself. Therefore, \( f(f(x)) = f\left(\frac{1}{x}\right) = \frac{1}{\left(\frac{1}{x}\right)} = x \). This shows the process of substitution for the same function repeatedly.

Key Concepts

Understanding Composite Functions

Composite functions are formed by applying one function to the result of another function. Imagine functions like machines: you input something, the machine does its magic, and produces an output. For composite functions, the output of one function becomes the input for another. This is often written as \(f(g(x))\) or \(g(f(x))\) where you substitute the inside function into the outer function.When dealing with composite functions:

• Identify the inner and outer functions. In \(f(g(x))\), \(g(x)\) is the inner function and \(f\) is the outer function.
• Substitute the output of the inner function into every occurrence of \(x\) in the outer function.
• Simplify the resulting expression if possible.

This concept is fundamental in calculus and aids in understanding how changes in one function affect another when combined.

Exploring Mathematical Functions

Mathematical functions define relationships between two variables, often x and y. A function expresses how one variable depends on another. For example, in \(f(x) = \frac{1}{x}\), each input \(x\) is paired with a single output \(\frac{1}{x}\). Functions can be linear, quadratic, polynomial, exponential, or even logarithmic.When exploring functions, consider their:

• Domain: All possible input values for the function. For \(f(x) = \frac{1}{x}\), \(x\) cannot be zero because division by zero is undefined.
• Range: All possible output values. For instance, \(g(x) = x^2 + 1\) will always produce numbers greater than or equal to 1.
• Graph: A visual representation that can help understand the function's behavior.

Functions are the backbone of calculus, allowing for the modeling of natural and scientific processes.

Calculus Problem Solving Techniques

Solving calculus problems often involves understanding how functions behave and interact with each other. Calculus provides tools like differentiation and integration to analyze these behaviors further. In the context of composite functions, solving problems involves substitution and simplification. Here are some steps to ease calculus problem solving:

• If dealing with composite functions, always substitute accurately and simplify step by step.
• Check the domain of each function before integrating them, to ensure no unintended errors occur.
• Use graphical representations to visualize the function's behavior, especially when dealing with complex expressions.

These techniques help break down complex calculus problems into manageable steps, aiding both beginners and advanced learners in problem-solving.