Problem 49 - 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. $$ \begin{aligned} &f(x)=x-1\\\ &\begin{array}{|r|l|} \hline {}{\underline{\phantom{xx}}}{\boldsymbol{c}} & \boldsymbol{f}(\boldsymbol{x})=\boldsymbol{x}-1 \\\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

The function values (f(x)) and corresponding ordered pairs are: -3 and (-2, -3), -2 and (-1, -2), -1 and (0, -1), 0 and (1, 0), 1 and (2, 1). These ordered pairs give us the graph of the function \(f(x) = x - 1\).

Step 1: Evaluate the function

To evaluate the function \(f(x) = x - 1\) for a specific value of \(x\), substitute the given \(x\) value into the expression and calculate the corresponding \(f(x)\) value. Do this for each provided \(x\) value: -2, -1, 0, 1, and 2.

Step 2: Represent the function values as ordered pairs

Now that we have the function values for each \(x\), represent them as ordered pairs (x, f(x)). Here are the pairs formed: (-2, -3), (-1, -2), (0, -1), (1, 0), (2, 1). These ordered pairs represent points on the graph of the function.

Step 3: Create the graph

Use these ordered pairs to graph the function. Since \(f(x) = x - 1\) is a linear function, the points should form a straight line when plotted. The x-axis represents the input (\(x\)) and the y-axis represents the output (\(f(x)\)). Plot the points and draw a line that passes through them. This line is the graph of the function.

Key Concepts

Function Evaluation

Function evaluation is all about finding the output of a function for specific input values. In this exercise, we're working with a linear function: \( f(x) = x - 1 \). To evaluate the function, follow these simple steps:

• Take the given value of \( x \)
• Substitute it into the function \( f(x) \)
• Calculate the result

For example, if \( x = -2 \), then substitute \( -2 \) into \( f(x) = x - 1 \) to get \( f(-2) = -2 - 1 = -3 \). Do this for each value of \( x \) provided: \(-2, -1, 0, 1,\) and \(2\).
This will give you the output for each input, making it easier to understand how the function behaves.

Ordered Pairs

Once you have calculated the function values, you can represent them using ordered pairs. These pairs take the form \((x, f(x))\), showing the relationship between the input \(x\) and output \(f(x)\).
Ordered pairs are essential because they allow you to see the function as a collection of "points" that can be plotted. From our example, after function evaluation, the ordered pairs would be:

• \((-2, -3)\)
• \((-1, -2)\)
• \((0, -1)\)
• \((1, 0)\)
• \((2, 1)\)

These pairs are fundamental in plotting the graph of the function and showing the linear relationship visually.

Graphing Functions

Graphing a function is the process of drawing its representative points on a coordinate plane and connecting them to reveal the shape of the function. With the ordered pairs in hand from function evaluation, you're ready to plot the points on a coordinate plane.
Follow these steps to graph \( f(x) = x - 1 \):

• Plot each ordered pair on the graph
• Make sure you correctly identify the \(x\) (horizontal) and \(f(x)\) or \(y\) (vertical) axes
• Once all points are plotted, draw a straight line through them

Since \( f(x) = x - 1 \) is a linear function, the points will align perfectly, forming a straight line. This graph represents the function visually and can help you understand how changing \(x\) affects \(f(x)\).

Coordinate System

The coordinate system is a grid that allows you to plot the position of points defined by ordered pairs. It's composed of a horizontal \(x\)-axis and a vertical \(y\)-axis (or \(f(x)\) in some contexts).
Each point on this plane is identified by an ordered pair \((x, y)\) indicating its horizontal and vertical positions, respectively.When graphing functions like \( f(x) = x - 1 \), it's crucial to understand that:

• The \(x\)-axis represents the input values of the function
• The \(y\)-axis or \(f(x)\)-axis represents the corresponding output values

By plotting the ordered pairs on this two-dimensional plane, you can visually explore the relationship between the inputs and outputs of the function, offering a clearer understanding of linear functions.