Problem 15 - Calculus for Biology and Medicine

Question

Suppose that $$ f(x)=1-x^{2}, \quad x \in \mathbf{R} $$ and $$ g(x)=2 x, \quad x \geq 0 $$ (a) Find $$ (f \circ g)(x) $$ together with its domain. (b) Find $$ (g \circ f)(x) $$ together with its domain.

Step-by-Step Solution

Answer

(f o g)(x) = 1 - 4x^2 with domain x ≥ 0; (g o f)(x) = 2 - 2x^2 with domain -1 ≤ x ≤ 1.

Step 1: Understand Function Composition

Function composition involves applying one function to the results of another. Given two functions, \( f(x) \) and \( g(x) \), the composition \( (f \circ g)(x) \) means we first apply \( g \), and then apply \( f \) to the result of \( g(x) \).

Step 2: Compute (f o g)(x)

To find \( (f \circ g)(x) \), we start by substituting \( g(x) = 2x \) into \( f(x) \). We have: \[ f(g(x)) = f(2x) = 1 - (2x)^2. \] Simplifying further gives: \[ f(g(x)) = 1 - 4x^2. \]

Step 3: Determine the Domain of (f o g)(x)

The domain of \( g(x) \) is \( x \geq 0 \) since \( g(x) = 2x \). The domain of the composition \( (f \circ g)(x) \) is the domain of \( g(x) \) restricted to where \( f(x) \) is defined, which, in this case, results in \( x \geq 0 \) because \( f(x) = 1 - x^2 \) is defined for all real numbers.

Step 4: Compute (g o f)(x)

Now we find \( (g \circ f)(x) \). Substitute \( f(x) = 1 - x^2 \) into \( g(x) \), which gives \[ g(f(x)) = g(1 - x^2) = 2(1 - x^2). \] Simplifying this expression yields: \[ g(f(x)) = 2 - 2x^2. \]

Step 5: Determine the Domain of (g o f)(x)

The function \( f(x) = 1 - x^2 \) is defined for all real numbers, but \( g(x) \) requires non-negative input, so we set \( 1-x^2 \geq 0 \). This leads to the inequality \( x^2 \leq 1 \), which implies \( -1 \leq x \leq 1 \). Therefore, the domain of \( (g \circ f)(x) \) is \([-1, 1]\).

Key Concepts

Domain of a Function

When dealing with functions, the domain is an essential concept to grasp. It refers to all the possible inputs (x-values) that a function can accept without causing issues like division by zero or taking the square root of a negative number. Functions such as polynomial functions, like the ones in the exercise, are typically defined for all real numbers unless specified otherwise.
For instance, the function \( f(x) = 1 - x^2 \) is defined for all real numbers. There are no constraints on \( x \), meaning anything you put in is valid. However, things change when we deal with \( g(x) = 2x \), which has a restriction of \( x \geq 0 \). This restriction is crucial when finding the domain of composites like \((f \circ g)(x)\).

• Always start by considering the domain of the inner function (like \( g(x) \) when calculating \((f \circ g)(x)\)).
• Ensure the results from the inner function fit into the domain of the outer function (\( f(x) \) in this case).
• Combine these restrictions to define the domain of the composite function.

Understanding domains is key to avoiding erroneous results and ensuring your functions work as intended.

Polynomial Function

Polynomial functions, like \( f(x) = 1 - x^2 \), form the building blocks of many mathematical problems. A polynomial is simply an expression made up of variables and coefficients, only using the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

• The degree of a polynomial is the highest exponent of the variable. For example, \( 1 - x^2 \) is a second-degree or quadratic polynomial.
• Polynomials are continuous and smooth, meaning they have no gaps or sharp points.
• They are defined for all real numbers, leading to a wide domain unless otherwise restricted by composition, as shown in the exercise.

Understanding the structure and properties of polynomial functions helps in identifying their behavior over different domains. In compositions, remembering these properties helps explain why combining functions affects their outputs and allowable inputs.

Inequalities

Inequalities play a pivotal role when determining the domain of a function, especially in compositions. They offer a way to define a range where a function is valid. For example, in the exercise, when creating \((g \circ f)(x)\), we establish an inequality to ensure the input to \( g \) is valid.
In these problems, setting \( 1-x^2 \geq 0 \), we imply finding \( x \) values where this statement holds true:

• Reorganize it to \( x^2 \leq 1 \).
• Take the square root of both sides, resulting in two inequalities: \( -1 \leq x \leq 1 \).
• These inequalities define the exact range of x-values permissible for our composite function.

Inequalities offer a simple method for identifying these critical ranges, making them indispensable in function analysis and ensuring the inputs are within the bound of each function's domain, avoiding undefined outputs.