Question

Graph a function whose domain is $$ \\{x \mid-3 \leq x \leq 8, \quad x \neq 5\\} $$ and whose range is $$ \\{y \mid-1 \leq y \leq 2, \quad y \neq 0\\} $$ What point(s) in the rectangle \(-3 \leq x \leq 8,-1 \leq y \leq 2\) cannot be on the graph? Compare your graph with those of other students. What differences do you see?

Short answer

Points on the graph cannot be at x=5 or y=0 within the given domain and range.

Step-by-step solution

Identify the Domain

Recognize that the domain of the function is given by \[-3 \leq x \leq 8, \quad x \eq 5\]. This means the x-values for the function range from -3 to 8, but the function is undefined at x = 5.

Identify the Range

Recognize that the range of the function is given by \[-1 \leq y \leq 2, \quad y \eq 0\]. This means the y-values for the function range from -1 to 2, but the function does not take the value y = 0.

Determine Missing Points in the Rectangle

Given the domain and range, identify the specific points that are excluded. The points where the function is undefined are at x = 5 and y = 0.

Plot the Rectangle

Now plot the rectangle defined by the given domain and range: o X-axis from -3 to 8o Y-axis from -1 to 2.

Identify Excluded Points

Mark the excluded points: any point that has x = 5 and any point that has y = 0. Especially, (5, y) for all y in the range, and (x, 0) for all x in the domain, are excluded.

Compare With Peers

Discuss and compare your graph with others. Look for how they handled the excluded points (x = 5 and y = 0) to note any differences or similarities.

Key concepts

Understanding the Function Domain

The domain of a function represents all the possible x-values that can be input into the function. In this exercise, the domain is given by \(-3 \leq x \leq 8, \quad x \eq 5\).
This means that the function takes x-values starting from -3 up to 8 but skips the value at x = 5.
To visualize this domain on the coordinate plane, plot the range of x-values from -3 to 8 along the x-axis.
Make sure not to include the value at x = 5 as the function is undefined at this point.
It’s like marking a specific segment on the x-axis but leaving out one point within that segment.
The interval from -3 to 8 includes all numbers in between, except exactly at 5.
On a graph, you’d often represent excluding a point with an open circle or a hole.

Grasping the Function Range

The range of a function reveals all the possible y-values that the function can output. Here, the range is given by \(-1 \leq y \leq 2, \quad y \eq 0\).
This defines the function to have y-values from -1 to 2 but it skips the value at y = 0.
To put this on the coordinate plane, you’d mark the y-values spanning -1 to 2 on the y-axis.
Keep in mind to leave out the point at y = 0 since the function does not generate this value.
On the graph, represent this exclusion with an open circle or a gap at y = 0.
The interval includes all values from -1 to 2 but excludes exactly 0.
This ensures we accurately show the function’s outputs within the specified range limit.

Plotting on the Coordinate Plane

The coordinate plane is a two-dimensional grid used to graphically represent functions.
It consists of two axes: the horizontal x-axis and the vertical y-axis. Each point on the plane represents a coordinate pair (x, y).
For graphing our function, the domain \(-3 \leq x \leq 8, \quad x \eq 5\) and range \(-1 \leq y \leq 2, \quad y \eq 0\) set the boundaries.
Begin by drawing a rectangle using the domain and range boundaries:
• X-axis from -3 to 8
• Y-axis from -1 to 2
This rectangle outlines the area where our function exists.
Next, mark the excluded points (x = 5 and y = 0). For every y-value, skip x = 5 by drawing an open circle along a vertical line at x = 5.
Meanwhile, skip y = 0 for every x-value by drawing open circles along a horizontal line at y = 0.
This shows that the function doesn’t pass through these lines.