Problem 31 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=|x|, g(x)=\sqrt{x}\), and \(h(x)=x-2\). Find the domain for each of the following. (a) \(j(x)=g(h(x))\) (b) \(k(x)=h(g(x))\)

Step-by-Step Solution

Answer

The domain of \(j(x) = g(h(x))\) is \(x \geq 2\) and the domain of \(k(x) = h(g(x))\) is \(x \geq 0\).

Step 1: Find the Domain of j(x) = g(h(x))

Let's find the domain of \(j(x) = g(h(x))\). Here, the inner function is \(h(x) = x - 2\) and the outer function is \(g(x) = \sqrt{x}\). The domain of \(h(x)\) is all real numbers, and the domain of \(g(x)\) is \(x \geq 0\), so the composite function restricts the domain to \((x - 2) \geq 0\). Solving this inequality results in \(x \geq 2\). So, the domain of \(j(x) = g(h(x))\) is \(x \geq 2\).

Step 2: Find the Domain of k(x) = h(g(x))

Let's find the domain of \(k(x) = h(g(x))\). Here, the inner function is \(g(x) = \sqrt{x}\) and the outer function is \(h(x) = x - 2\). The domain of \(g(x)\) is \(x \geq 0\), and since \(h(x)\) accepts all real numbers, the domain of \(k(x)\) is the same as the domain of the inner function. So, the domain of \(k(x) = h(g(x))\) is \(x \geq 0\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial since it tells us all the possible input values a function can accept. Simply put, the domain consists of all the values for which the function is defined. Let's dive a little deeper into this concept.

When examining the domain, we need to check all the operations and definitions involved in the function. This involves:

• Checking for division by zero.
• Identifying values that lead to negative numbers under a square root, since square roots of negative numbers are not defined in the real number system.

In our exercise, the function \( g(x) = \sqrt{x} \) implies that \( x \) must be greater than or equal to zero—\( x \geq 0 \)—to stay within the realm of real numbers.

When we assess composite functions like \( j(x) = g(h(x)) \), we analyze both \( h(x) = x - 2 \) and \( g(x) \). The domain of \( h(x) \) is all real numbers, but \( j(x) \)'s domain is determined by\( g(x) \). We find \( x - 2 \geq 0 \), resulting in \( x \geq 2 \). So, the domain is restricted to values of \( x \) where \( x \) is at least 2.

Composite Functions

Composite functions involve the combination of two functions, where the output of one function feeds into the input of another. The notation \( (f \circ g)(x) \, \) or \( f(g(x)) \), represents this combination. Each function is like a mini-machine that processes numbers, and when they are combined, one function's output is automatically used by the next.

For composite functions, it's crucial to focus on the domains, not just for the individual functions, but also their combination. The domain of a composite function involves checking the domain of both the inner and outer functions.

• Ensure the output from the inner function can be an input to the outer function.
• Identify any additional restrictions posed by the composite structure.

In the exercise,\( j(x) = g(h(x)) \) demonstrates composite functioning. Here, \(h(x) = x - 2\)'s output directly impacts \(g(x) = \sqrt{x} \). The domain for \( g \), \( x \geq 0 \), limits the valid inputs for the function \( j(x) \), resulting in \( x \geq 2 \) in this scenario.

Piecewise Functions

Piecewise functions are not activated for this particular exercise, but understanding them enhances our comprehension of composite function behavior. These functions are special since they apply different rules for different parts of their domain. Imagine a function behaving like function A for some inputs, and like function B for others. It’s like choosing the best route on a map depending on traffic, time, or other conditions.

Here's how to recognize and solve them fairly simply:

• Identify the specific conditions or intervals where each piece of the function applies.
• Analyze the domain for each interval, ensuring there are no overlaps or gaps.
• Understand how each piece of the function contributes to the overall output.

While piecewise functions seem complicated, they help articulate complex behaviors, and each piece is treated individually, sometimes resembling the tasks we performed with composite functions. This methodical approach ensures that we're looking at each piece in its own context while understanding how they collectively portray the entire function.