Problem 17 - Calculus for Biology and Medicine

Question

Suppose that $$ f(x)=3 x^{2}, \quad x \geq 3 $$ and $$ g(x)=\sqrt{x}, \quad x \geq 0 $$ Find \((f \circ g)(x)\) together with its domain.

Step-by-Step Solution

Answer

\((f \circ g)(x) = 3x\) with domain \(x \geq 9\).

Step 1: Understand the Composition Function

The composition of two functions, \((f \circ g)(x)\), means that you substitute the output of function \(g\) as the input of function \(f\). This means \(f(g(x))\).

Step 2: Substitute \(g(x)\) into \(f(x)\)

The function \(g(x)\) is given as \(\sqrt{x}\). Substitute \(g(x)\) into \(f(x) = 3x^{2}\) to get \(f(g(x)) = 3(\sqrt{x})^{2}\).

Step 3: Simplify the Expression

Since \((\sqrt{x})^{2} = x\), substitute it back into the function: \(f(g(x)) = 3x\). Thus, \((f \circ g)(x) = 3x\).

Step 4: Determine the Domain

The domain of \(f(g(x))\) depends on the domains of both \(f(x)\) and \(g(x)\). Since \(g(x) = \sqrt{x}\) requires \(x \geq 0\) and \(f(x)\) has a domain requirement \(x \geq 3\), the composition \(f(g(x)) = 3x\) requires \(g(x) \geq 3\). Hence, \(x \geq 9\), since \(\sqrt{x} \geq 3\) implies \(x \geq 9\).

Key Concepts

Domain of a Function

In mathematics, the domain of a function is the set of all possible input values (usually denoted as 'x') which will result in a valid output value. For example, if a function is defined by \(f(x) = 1/x\), we cannot input \(x = 0\) because division by zero is undefined. Rather, the domain of \(f(x)\) excludes zero.
The domain can vary based on the structure of the function. To find the domain, you need to consider factors like:

• The mathematical operations involved (e.g., division, square roots)
• Constraints or restrictions given explicitly in the function definition
• Practical considerations influencing valid input values

In our exercise, understanding the domain of both \(f(x)\) and \(g(x)\) is crucial before performing function composition. We know from the exercise:

• For \(f(x) = 3x^2\), the domain is \(x \geq 3\).
• For \(g(x) = \sqrt{x}\), the domain is \(x \geq 0\).

When these two functions are composed together, the resulting domain must satisfy both conditions. This computation yields that the domain of \((f \circ g)(x)\) is such that \(g(x) \geq 3\). Solving this inequality tells us \(x \geq 9\).

Function Substitution

Function substitution is an essential concept used to evaluate composite functions. When substituting, you use the entire output of one function as an input into another. This is what happens in function composition.
Suppose you have two functions: \(f(x)\) and \(g(x)\), and you want to find the composition \((f \circ g)(x)\). This composition implies that wherever there's an 'x' in \(f(x)\), you replace it with \(g(x)\). Let's break it down:

• First, calculate \(g(x)\).
• Take this result and substitute it wherever 'x' appears in \(f(x)\).

For our exercise, \(g(x) = \sqrt{x}\). Substituting this into \(f(x) = 3x^2\) gives \(f(g(x)) = 3(\sqrt{x})^2\).
After calculation, you'll notice that \((\sqrt{x})^2 = x\), thus the simplified form is \(3x\). The method of function substitution eases the computation process and is a powerful tool in finding composite functions.

Square Root Function

The square root function, symbolized as \(\sqrt{x}\), is a widely used mathematical function. It specifies that given a number \(x\), the output is a number \(y\) such that \(y^2 = x\). Essentially, it "undoes" squaring.
When we work with square roots, considering the domain is crucial to avoid undefined or impractical outputs.

• The square root function is only real and defined for non-negative numbers. This means \(x \geq 0\).
• Inside a composition, any constraints from other functions must also be respected.

In our example, \(g(x) = \sqrt{x}\) has a domain of \(x \geq 0\) intrinsically. When combined with the domain condition from \(f(x)\), these mutual influences help us determine the domain for the composite function \((f \circ g)(x)\). In a more practical view, this interpretation ensures all mathematical expressions involved result in real numbers and valid computations.