Problem 15 - Calculus for Biology and Medicine

Question

Suppose that \(f(x)=1-x, x \in \mathbf{R}\), and \(g(x)=\sqrt{x}, x \geq 0\). (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 ∘ g)(x) = 1 - \sqrt{x}, domain [0, ∞); (g ∘ f)(x) = \sqrt{1 - x}, domain (-∞, 1].

Step 1: Define Composite Function (f ∘ g)(x)

The composite function \((f \circ g)(x)\) means substituting \(g(x)\) into \(f(x)\). Given \(f(x) = 1 - x\) and \(g(x) = \sqrt{x}\), we have \((f \circ g)(x) = f(g(x)) = f(\sqrt{x}) = 1 - \sqrt{x}\).

Step 2: Determine Domain of (f ∘ g)(x)

The domain of \( (f \circ g)(x) = 1 - \sqrt{x} \) is restricted by \(g(x) = \sqrt{x}\), which requires \(x \geq 0\) to ensure the square root is defined. So, the domain of \((f \circ g)(x)\) is \([0, \infty)\).

Step 3: Define Composite Function (g ∘ f)(x)

The composite function \((g \circ f)(x)\) means substituting \(f(x)\) into \(g(x)\). Given \(g(x) = \sqrt{x}\) and \(f(x) = 1 - x\), we have \((g \circ f)(x) = g(f(x)) = g(1 - x) = \sqrt{1 - x}\).

Step 4: Determine Domain of (g ∘ f)(x)

The domain of \((g \circ f)(x) = \sqrt{1 - x}\) requires \(1 - x \geq 0\) to ensure the square root is valid. Solving this inequality, we get \(x \leq 1\). Therefore, the domain is \((-fty, 1]\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (typically represented by the variable \(x\)) that can be used without causing mathematical complications, like division by zero or taking the square root of a negative number.
In essence, the domain is a map of where the function is defined. To find the domain, consider the expression inside the function. You need to ensure that any mathematical operations within this function are possible. For instance, when dealing with square root functions, you must have non-negative values under the square root sign to avoid undefined scenarios.
The domain is often represented in interval notation, such as \([0, \infty)\), indicating that \(x\) can be any number from 0 to infinity, inclusive of 0. When determining the domain of composite functions like

• \((f \circ g)(x)\), individual domains of \(f(x)\) and \(g(x)\) should be considered.
• This is because the resulting composition must meet the criteria of both functions involved.

Function Composition

Function composition involves creating a new function by applying one function to the results of another function. If you have two functions, say \(f(x)\) and \(g(x)\), the composition

• \((f \circ g)(x)\) means you first apply \(g(x)\) and then use the output as an input for \(f(x)\).
• Essentially, you're nesting one function inside another like a Russian doll.

The order of composition matters and switching the order of \(f\) and \(g\) can yield different results

• For instance, \((f \circ g)(x) = f(g(x))\) might differ significantly from \((g \circ f)(x) = g(f(x))\).

This happens because the function's nature changes based on what operation occurs first.To solve for a composite function, you substitute the entirety of one function into the other. Using our example from above, with \((f \circ g)(x) = f(\sqrt{x}) = 1 - \sqrt{x}\), you can observe that applying \(g(x)\) first and then \(f(x)\) results in a distinct composition.
The resulting domain of the composite function is determined based on these substitutions.

Square Root Function

A square root function is a function of the form \(g(x) = \sqrt{x}\). This function finds the square root of \(x\). A critical aspect of square root functions is the restriction on the domain.
The square root of a number is only defined when that number is non-negative, that is, zero or positive. Thus, the domain of the function \(g(x) = \sqrt{x}\) is \([0, \infty)\), meaning it includes 0 and all positive numbers.
When we compose functions involving square root operations, it is essential to maintain this constraint. The square root operation reduces the number of eligible inputs, enforcing that values within the function cannot turn negative.
The composite function, like \((g \circ f)(x) = \sqrt{1 - x}\), must check that \(1 - x \geq 0\). Solving this inequality helps confirm which values of \(x\) keep the expression under the square root non-negative.

• This results in the composite domain \((-\infty, 1]\) to ensure a real number output for the operation.

By carefully analyzing both the expression under the root and the output, we ensure that the square root function maintains its integrity in compositions.