Problem 60 - Calculus: One and Several Variables

Question

Use a graphing utility to draw several views of the graph of the function. Select the one that most accurately shows the important features of the graph. Give the domain and range of the function. $$f(x)=\sqrt{x^{3}-8}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\sqrt{x^{3}-8}\) starts at \(x=2\) and goes upwards as \(x\) increases. Thus, the domain is \([2,+\infty)\) and the range is \([0,+\infty)\).

Step 1: Graph the function.

First of all, use a graphing utility to plot the function \(f(x)=\sqrt{x^{3}-8}\). Observe that the graph exists only for the values of \(x\) where the expression under the square root is zero or positive. It means \(x>=2\) as \(x^{3}-8 = 0\) gives \(x = 2\).

Step 2: Find the domain

From the graph and the observation made in step 1, the domain of the function \(f(x)=\sqrt{x^{3}-8}\) can be determined. The domain is the set of all possible x-values for which the function is defined. In this case, that would be all the values of \(x\) which are greater than or equal to 2. So, domain is \([2,+\infty)\).

Step 3: Find the range

The range of a function is the set of all possible y-values of the function. Looking at the graph of the function \(f(x)=\sqrt{x^{3}-8}\), you'd notice that the graph starts from y=0 (when \(x = 2\)) and goes upward for higher values of \(x\). Hence, the range of the function is \([0, +\infty)\)

Key Concepts

Domain of a Function

The domain of a function is essentially the collection of all input values (or x-values) for which the function produces real output values. In simpler terms, imagine the domain as the set of all possible choices you can plug into the function without making it undefined. For instance, in the function \(f(x) = \sqrt{x^3 - 8}\), we must ensure that values inside the square root are non-negative because you cannot take the square root of a negative number in the real number system. This is why we solve the equation \(x^3 - 8 \geq 0\) to find the domain. Solving gives \(x \geq 2\). Therefore, any value greater than or equal to 2 can be input into the function, making the domain \([2, +\infty)\). In practical terms, when you're graphing, imagine slicing off all the x-values to the left of 2, because they don’t make sense for the function. This slice forms the framework of the domain.

Range of a Function

The range of a function is the set of all feasible output values (y-values) that the function can deliver. Think of the range as what comes out after you have fed the domain into the function. In our function \(f(x) = \sqrt{x^3 - 8}\), once the domain is set, you can examine how the y-values or outputs behave. When \(x = 2\), the smallest value in the domain, the output is \(f(2) = \sqrt{2^3 - 8} = \sqrt{0} = 0\). As x increases, the value under the square root increases, and so does the value of \(f(x)\). Therefore, the range is all y-values from 0 upwards, which can be represented as \([0, +\infty)\). This means the function graph stretches upwards from y = 0 with no ceiling – it can take on any, higher y-value as x increases.

Square Root Functions

Square root functions are a special type of function where the variable is under a square root sign. Graphically, these functions showcase a smooth, gradual curve. A simple example is \(f(x) = \sqrt{x}\), where the domain is \([0, +\infty)\) and its range is also \([0, +\infty)\). For the function \(f(x) = \sqrt{x^3 - 8}\), notice how the variable term inside the square root is more complex. It affects both the domain and the appearance of the graph.
Key characteristics of square root functions include:

• They start from a particular point or curve upwards continually if defined appropriately (for positive x-values).
• They can be horizontally shaped, depending on the function inside the root.
• Their domain needs careful determination to avoid undefined outputs.

In such functions, part of the journey is understanding the setup under the square root and how it impacts the function's domain and range.