Problem 10 - Applied Calculus

Question

Find the following: (a) \(f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=x^{2}+1\( and \)g(x)=\ln x$$

Step-by-Step Solution

Answer

(a) \((\ln x)^2 + 1\); (b) \(\ln(x^2 + 1)\); (c) \(x^4 + 2x^2 + 2\)."}

Step 1: Understand the Functions

Let's start by defining what each function does: For the function \( f(x) = x^2 + 1 \), \( f(x) \) takes any real number \( x \), squares it, and then adds 1 to the result. For \( g(x) = \ln x \), \( g(x) \) takes a positive real number \( x \) and returns its natural logarithm.

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

To find \( f(g(x)) \), substitute \( g(x) = \ln x \) into \( f(x) \). This gives us \( f(g(x)) = f(\ln x) = (\ln x)^2 + 1 \). Thus, \( f(g(x)) = (\ln x)^2 + 1 \).

Step 3: Calculate \( g(f(x)) \)

For \( g(f(x)) \), substitute \( f(x) = x^2 + 1 \) into \( g(x) \). This results in \( g(f(x)) = g(x^2 + 1) = \ln(x^2 + 1) \). Hence, \( g(f(x)) = \ln(x^2 + 1) \).

Step 4: Determine \( f(f(x)) \)

To find \( f(f(x)) \), substitute \( f(x) = x^2 + 1 \) back into itself. This yields \( f(f(x)) = f(x^2 + 1) = (x^2 + 1)^2 + 1 \). Simplify this to get \( (x^2 + 1)^2 = x^4 + 2x^2 + 1 \), so \( f(f(x)) = x^4 + 2x^2 + 1 + 1 = x^4 + 2x^2 + 2 \).

Key Concepts

Function Composition

Function composition is a powerful mathematical tool that allows you to build more complex functions from simpler ones. Think of it like a process where one function is evaluated and then its output becomes the input for another function. This is frequently written as \(f(g(x))\) or \(g(f(x))\). Function composition is essential in numerous areas of mathematics because it helps to simplify complex problems and analyze multi-step calculations.

Here's a simple guide on how to approach function composition:

• Identify each function individually. For example, if you have \(f(x) = x^2 + 1\) and \(g(x) = \ln x\), understand what each function does.
• Determine the order of operations. In \(f(g(x))\), calculate \(g(x)\) first and then use its result as the input for \(f(x)\).
• Substitute accordingly. For \(g(f(x))\), substitute \(f(x)\) into \(g(x)\) to get the final form of the composed function.

Understanding function composition can greatly improve problem-solving skills because it teaches you to think in terms of processes rather than separate actions.

Natural Logarithm

The natural logarithm, represented as \(\ln x\), is a logarithm to the base \(e\), where \(e\) is approximately 2.718. It holds a special place in mathematics due to its appearances in calculus, real analysis, and complex systems.

Some key points about natural logarithms include:

• They are only defined for positive real numbers, as you cannot take the logarithm of zero or negative numbers.
• The value of \(\ln 1\) is 0 because \(e^0=1\).
• Natural logarithms are useful for solving equations involving exponentiation, such as finding the time it takes for an investment to grow given a fixed interest rate compounded continuously.

In the context of function composition, it's important to remember the domain restrictions of the natural logarithm. For example, in \(f(g(x)) = (\ln x)^2 + 1\), \(x\) must be positive, otherwise, the natural logarithm does not exist, and thus the composition is undefined.

Polynomial Functions

Polynomial functions are mathematical expressions of the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a_i\) is a constant coefficient, and \(n\) is a non-negative integer. These functions are very common and cover a broad range of equations from simple quadratic functions to complex higher-order polynomials.

For example, the function \(f(x) = x^2 + 1\) is a quadratic polynomial, where each term consists of a coefficient and a power of \(x\).

• Quadratic polynomials have the general form \(ax^2 + bx + c\), and their graphs are parabolas.
• Higher degree polynomials can have more twists and turns, and their behavior is determined by the leading term, \(a_nx^n\).
• Adding or composing polynomials creates new polynomials, as seen in \(f(f(x)) = x^4 + 2x^2 + 2\), showcasing how functions of this type combine and expand.

When dealing with polynomial functions, it is crucial to simplify them correctly and understand how their various terms contribute to their overall form and graph. This understanding can aid in making predictions about the function's behavior and solving real-world problems.