Use a graph to determine whether the given function is continuous on its domain. HINT [See Example 1.]
f(x) =
x + 7 if x < 0
2x − 5 if x ≥ 0
1
continuous discontinuous
If it is not continuous on its domain, list the points of discontinuity. (If there is no such point, enter NONE.)
x = 2
In google type: "function graphs online"
When you see list buf results click on:
rechneronline.de/function-graphs/
When page be open in blue rectacangle
type: x+7
in grey rectacangle click: 2*x-5
In Display properties set:
Range x-axis -20 to 20
and
Range y-axis -20 to 20
Then click option Draw
Continuous
To determine whether the given function is continuous on its domain, we need to plot its graph.
1. For x < 0, the function is defined as f(x) = x + 7. This is a linear function with a positive slope of 1 and a y-intercept of 7.
2. For x ≥ 0, the function is defined as f(x) = 2x − 5. This is also a linear function with a positive slope of 2 and a y-intercept of -5.
On the graph, we can plot these two linear functions separately for their respective domains.
For x < 0, the graph will be a straight line with a positive slope starting from the point (0, 7) extending towards the left side of the y-axis.
For x ≥ 0, the graph will be a straight line with a positive slope starting from the point (0, -5) extending towards the right side of the y-axis.
Next, we need to consider the point of discontinuity, which is x = 0. At x = 0, the function transitions from the line with slope 1 (for x < 0) to the line with slope 2 (for x ≥ 0).
On the graph, when x approaches 0 from the left side (x < 0), the value of f(x) approaches 0 + 7 = 7. So, there is a hole in the graph at the point (0, 7).
When x approaches 0 from the right side (x ≥ 0), the value of f(x) approaches 2(0) - 5 = -5. So, the graph continues smoothly on the line at the point (0, -5).
To summarize,
- The graph of the function consists of two lines with a hole at x = 0.
- The function is discontinuous at x = 0.
- The points of discontinuity are (0, 7).
Therefore, the given function f(x) is discontinuous on its domain at x = 0.
This is very simple and quick to graph.
Have you tried to graph these?
What don't you understand?