Problem 35 - Calculus: One and Several Variables

Question

Give the domain and range of the function. $$f(x)=\sqrt{x-4}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\sqrt{x-4}\) is \([4, +\infty)\) and the range is \([0, +\infty)\).

Step 1: Identify the argument of the square root function

The argument of the square root function is \(x-4\). We cannot take the square root of a negative value, so \(x-4\) must be greater than or equal to 0. This gives the inequality \(x-4 \geq 0\).

Step 2: Solve the inequality for the domain

Solving the inequality will give the domain of the function. To solve it, add 4 to both sides of the inequality \(x-4 \geq 0\), which yields \(x \geq 4\). Therefore, the domain of \(f(x)=\sqrt{x-4}\) is \([4, +\infty)\).

Step 3: Determine the range

The square root function always outputs values from 0 upwards when the radicand is non-negative. Therefore, the range of \(f(x)=\sqrt{x-4}\) is \([0, +\infty)\).

Key Concepts

Solving Inequalities

Solving inequalities is similar to solving equations, but instead of equal signs, we use inequality symbols to compare two values. These symbols include greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤).

When solving an inequality, we perform operations on both sides, just like with an equation, but we need to be careful—multiplying or dividing by a negative number will reverse the inequality. For example, solving the inequality from the exercise step-by-step, we start with \(x - 4 \geq 0\) and isolate \(x\) by adding 4 to both sides, resulting in \(x \geq 4\). It is crucial to plot the solutions on a number line or use interval notation, like \([4, +\infty)\), to visually represent the range of values that \(x\) can take.

Domain of a Function

The domain of a function includes all the possible inputs, or \(x\)-values, that the function can accept without violation of mathematical rules. For functions involving square roots, like \(f(x) = \sqrt{x - 4}\), the domain is constrained to prevent taking the square root of a negative number—since that's not defined within the set of real numbers.

In our exercise, the argument of the square root, \(x - 4\), must be non-negative. So we set up an inequality \(x - 4 \geq 0\) and solve for \(x\) to find the domain. After solving, we find that the domain of \(f(x)\) is \([4, +\infty)\), meaning \(f(x)\) can accept any real number greater than or equal to 4 according to the square root properties.

Range of a Function

The range of a function consists of all possible outputs, or \(y\)-values, that the function can produce. Unlike the domain, which is determined by the inputs, the range is impacted by the function's operations on those inputs.

For the square root function \(f(x) = \sqrt{x - 4}\) from our exercise, we start by inspecting the lowest value of the square root expression—since square roots produce non-negative outputs, the smallest value for \(f(x)\) is 0, when \(x - 4\) is 0 (or when \(x\) is 4). As \(x\) increases, \(f(x)\) also increases without bound. Thus, the range is \([0, +\infty)\). This tells us that the square root function's range starts from 0 and stretches indefinitely upwards.

Square Root Properties

Square root properties are essential for understanding functions that include square root operations, such as \(f(x) = \sqrt{x - 4}\). The square root function, denoted as \(\sqrt{\cdot}\), only accepts non-negative radicands—the numbers under the root symbol—since the square of any real number is non-negative. This leads to two key properties:

• The result of a square root is always non-negative, which defines the range of square root functions.
• The domain is limited to non-negative radicands to avoid undefined or complex results within the real number system.

When given a function like \(f(x)\), we utilize these properties to determine the function's domain and range. It's also worth noting that the square root of a perfect square (e.g., \(\sqrt{9} = 3\)) will be an integer, while other numbers may result in irrational outcomes.