Problem 13 - Calculus for Biology and Medicine

Question

Suppose that $$ f(x)=x^{2}, \quad x \in \mathbf{R} $$ and $$ g(x)=3+x, \quad x \in \mathbf{R} $$ (a) Show that $$ (f \circ g)(x)=(3+x)^{2}, \quad x \in \mathbf{R} $$ (b) Show that $$ (g \circ f)(x)=3+x^{2}, \quad x \in \mathbf{R} $$

Step-by-Step Solution

Answer

(f \circ g)(x) = (3 + x)^2, (g \circ f)(x) = 3 + x^2.

Step 1: Understanding the Composition of Functions

To solve this problem, we need to understand the concept of function composition. The composition \((f \circ g)(x)\) means that we apply function \(g\) first, and then apply function \(f\) to the result. Similarly, \((g \circ f)(x)\) means we apply function \(f\) first, and then apply \(g\) to the result.

Step 1: Compute \((f \circ g)(x)\)

To find \((f \circ g)(x)\), we substitute \(g(x) = 3 + x\) into \(f\). So, we need to evaluate \(f(g(x)) = f(3 + x)\). The function \(f\) is given as \(f(x) = x^2\), which means \(f(3 + x) = (3 + x)^2\). This verifies that\[(f \circ g)(x) = (3 + x)^2, \quad x \in \mathbf{R} \]

Step 2: Compute \((g \circ f)(x)\)

To find \((g \circ f)(x)\), we substitute \(f(x) = x^2\) into \(g\). So, we need to evaluate \(g(f(x)) = g(x^2)\). The function \(g\) is given as \(g(x) = 3 + x\), so \(g(x^2) = 3 + x^2\). This verifies that\[(g \circ f)(x) = 3 + x^2, \quad x \in \mathbf{R} \]

Key Concepts

Understanding Mathematical Functions

Mathematical functions are essential building blocks in algebra. A function defines a relationship where each input is related to exactly one output. Think of it like a machine; you enter a number, the machine applies some operations, and outputs a result.
For example, the function \( f(x) = x^2 \) takes each input value \( x \) from the real numbers \( \mathbf{R} \) and squares it. If you input 2, the output would be 4, since \( 2^2 = 4 \). Similarly, function \( g(x) = 3 + x \) adds 3 to any real number \( x \). Both functions serve different purposes and transform inputs in distinct ways.

• \( f(x) = x^2 \) is a basic quadratic function.
• \( g(x) = 3 + x \) is a linear function, adding a constant 3 to the input.

Understanding these functions individually helps grasp what happens when we combine them through composition.

Exploring Composite Functions

Composite functions are formed by combining two functions in a specific order. If we have functions \( f \) and \( g \), then the composite function \((f \circ g)(x)\) means we apply \( g \) first, then \( f \) to the result of \( g \). This composition is like a two-step process:

• First, compute \( g(x) \).
• Then, apply \( f \) to that result.

In our exercise, \((f \circ g)(x) = f(g(x)) = (3 + x)^2\), because after calculating \( g(x) = 3 + x \), we square the result when we apply \( f \). Similarly, the reverse composition \((g \circ f)(x)\) applies \( f \) first to get \( x^2 \), then \( g \) adds 3. This gives \( g(x^2) = 3 + x^2 \).
Composite functions can be powerful tools to express complex transformations that single functions cannot easily achieve. Understanding their processes greatly enhances problem-solving skills!

Applying Algebraic Operations in Composition

Algebraic operations are integral in evaluating composite functions. These operations allow us to manipulate expressions to understand function behavior better. When dealing with composites like \((f \circ g)(x)\), algebraic operations involve:

• Substituting expressions: Substitute the function \( g(x) = 3 + x \) into \( f \) to get \( (3 + x)^2 \).
• Simplifying expressions: Expand \( (3 + x)^2 \) to \( 9 + 6x + x^2 \) if needed for further analysis.

In our exercise, the calculation of \((g \circ f)(x)\) involves substitution and addition, resulting in \( 3 + x^2 \). These operations help illustrate the transformation process as they simplify different stages of computation.
Mastering algebraic operations allows for smoother calculations and deeper understanding of how different functions work together.