Question

Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. $$ f(x)=\frac{1}{2} x^{3}+2 $$

Short answer

The domain of the function \( f(x) = \frac{1}{2}x^3 + 2 \) is \( (-\infty, \infty) \) and the range is also \( (-\infty, \infty) \). The function graph looks like the graph of \( x^3 \), but stretched vertically by factor 1/2 and shifted 2 units up.

Step-by-step solution

Understand function properties

This is a cubic function, where \(x\) is raised to the power of 3. Cubic functions are always continuous and differentiable everywhere, which means their domain is all real numbers, or \( (-\infty, \infty) \).

Sketch the function graph

To sketch the function, first draw a standard cubic function \( y=x^3 \) which starts from the third quadrant, passes through the origin and goes to the first quadrant. Notice that as \( x \) becomes more negative or positive, the value of the function increases/decreases rapidly due to the cubic power. The function \( f(x)=\frac{1}{2}x^3 + 2 \) looks like a stretched version of the standard cubic function shifted 2 units up. It still passes through the point (0, 2) instead of the origin.

Find the range

The range of the function is the set of all possible output values (y-values). Considering the nature of this function, as \( x \) goes to \( -\infty \), \( y \) also goes to \( -\infty \), and as \( x \) moves to \( \infty \), \( y \) goes to \( \infty \). This means that the range is also all real numbers, or \( (-\infty, \infty) \).

Confirm using graphing utility

Finally, confirm the shape of this graph, the domain, and the range by using a graphing utility. The results should match your manually sketched graph and calculated domain and range.

Key concepts

Cubic Function Properties

A fundamental aspect of algebra involves understanding the characteristics of various functions, particularly cubic functions. These functions are defined by an equation where the variable is raised to the third power, taking the form of \( f(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants and \( a \) is non-zero. One of the most defining properties of cubic functions is their shape; they have an 'S' or 'snake' like curve that can either go from the top-left to bottom-right or vice versa, depending on the sign of the leading coefficient.

Another essential property is that cubic functions are continuous and differentiable for all real values. This implies they have no sharp corners or breaks. The end behavior of these functions is such that as \( x \) approaches positive or negative infinity, the function's value also tends towards infinity or negative infinity, respectively. Due to their nature, they always have a domain of \( (-fty, fty) \), which means they are defined for all real numbers.

Finding Domain and Range

The domain of a function is the complete set of possible values of the independent variable, in this case \( x \), for which the function is defined. As touched upon in the properties of cubic functions, the domain for any cubic function is \( (-fty, fty) \) because no matter what real value you substitute for \( x \), you will get a real number output.

On the other hand, the range can initially seem more complex. It refers to the set of possible output values, which are dependent on the function's formula and thus the values of \( x \). However, for cubic functions, the range is also \( (-fty, fty) \) because no matter what real number you choose for \( x \), whether it's immensely high or low, the cubic term will dominate, and the function can produce any real value as \( y \). Consequently, the task of sketching the graph of the function revolves around understanding these infinite bounds and the general shape of the cubic function.

Using Graphing Utilities

Graphing utilities are indispensable tools for visualizing and validating the characteristics of functions, particularly when dealing with complex functions like cubics. They allow for a visual representation that can be incredibly accurate, aiding in both understanding the conceptual underpinnings of the function and in verifying its properties. To employ a graphing utility, input the function's equation and observe the resulting graph.

The utility should display a graph that confirms the properties we've discussed: the 'S' shaped curve, the absence of breaks, and the correct intercepts. For the given exercise, we would input the function \( f(x)=\frac{1}{2} x^{3}+2 \) into the utility. The resulting graph should show the function's stretch in comparison to the standard \( y=x^3 \) graph and the upward shift by 2 units. It's a good practice to analyze the output rigorously, checking that the domain and range match our theoretical understanding. Using graphing utilities not only reinforces our grasp of these functions but also provides a powerful visual learning aid.