Problem 53 - Thinking Mathematically

Question

Evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=x^{3}+1\\\ &\begin{array}{|r|c|} \hline {}{\underline{\phantom{xx}}}{x} & f(x)=x^{3}+1 \\ \hline-3 & \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

The function values are \(f(-3)=-26\), \(f(-2)=-7\), \(f(-1)=0\), \(f(0)=1\), and \(f(1)=2\). These values are used to graph the function \(f(x)=x^{3}+1\). In the graph, the curve ascends from left to right, reflecting the nature of a cubic function with a positive leading coefficient.

Step 1: Evaluate Function for Given Values

Use the provided x-values to evaluate \(f(x)=x^{3}+1\) by substituting each x-value into the function. For example, when x=-3, \(f(x)=(-3)^{3}+1\), which equals -26. Perform this operation for all the given x-values.

Step 2: Create a New Table

Put the results from the previous step into a new table of ordered pairs:\[\begin{array}{|r|c|}\hline {}{x} & f(x)=x^{3}+1 \ \hline-3 & -26 \ \hline-2 & -7 \ \hline-1 & 0 \ \hline 0 & 1 \ \hline 1 & 2 \ \hline \end{array}\]In this table, each x-value is paired with its corresponding function value.

Step 3: Graph the Function

Use the ordered pairs to plot the graph of the function. For each pair \((x, f(x))\), use x as the x-coordinate and \(f(x)\) as the y-coordinate. Once all points are plotted, draw a smooth curve through the points to show the full function.

Key Concepts

Evaluate Functions

Understanding how to evaluate functions is a fundamental skill in algebra and calculus. Essentially, it involves substituting a given value into a function and performing the necessary arithmetic operations. For example, consider the function given in our exercise, f(x) = x^3 + 1. When asked to evaluate this function for x = -3, you replace the x in the function with -3. You then calculate (-3)^3 + 1, which simplifies to -27 + 1 or -26.

By following this process, you can find out what the function 'outputs' for each 'input' value of x. This is essential for creating a table of values and ultimately graphing the function, as it gives you specific points through which the graph will pass. It also introduces the concept of domain and range; here the domain is the set of all permissible values for x, and the range is the set of all possible outputs f(x).

Function Notation

Function notation is a way of writing functions that makes it clear which variables are independent (normally x) and which are dependent (normally f(x)). In the given function, f(x) denotes the dependent variable — the output of the function — and x is the independent variable, the input.

This concise notation helps us to easily see how the output is derived from the input, and is a universally accepted way of communicating mathematical functions. Remember, the notation does not only apply to simple algebraic expressions. It can represent anything from simple constants to complex, multi-variable algebraic or transcendental functions.

Ordered Pairs

Ordered pairs are the foundation of graphing on a Cartesian plane. These come in the form (x, y), where x is the first element denoting the horizontal position from the origin, and y is the second element indicating the vertical position. The function value f(x) usually represents the y value of the ordered pair.

In graphing, we use these coordinates to pinpoint precise locations on the plane. For the function in our exercise, after evaluating f(x) for each x, we combine them into ordered pairs such as (-3, -26), which tells us that the point lies three units to the left and twenty-six units down from the origin. When plotted, these pairs reveal the shape and direction of a function's graph.

Function Graphing

Graphing a function involves plotting the ordered pairs obtained from the function's table of values onto a Cartesian plane and connecting them with a curve that represents all the infinite points of the function. As you plot points like (-3, -26), (-2, -7), (-1, 0), (0, 1), and (1, 2), it's important to remember that these are just a few specific examples of the function's output.

The process of joining the points should reflect the continuous nature of the polynomial function, creating a smooth curve and avoiding sharp corners unless a corner is a feature of the function's graph. In the end, the graph visually represents the relationship between x and f(x), allowing us to quickly ascertain various function properties such as increasing/decreasing behavior, intercepts, and potential turning points.