Given the following piecewise function
f(x)= {-x+1 for x < 0
{-1 for 0 ≤ x ≤ 3
{-2x for x > 3
a) Find the domain
b) Find the range
c) Find the intercepts
d) Is f continuous on its domain? If not, state where f is discontinuous.
e) Graph the function
To find the domain of the function, we need to see where the function is defined. Looking at the given piecewise function, we have three different equations for three different intervals.
a) Finding the domain:
- The first equation, f(x) = -x+1, is defined for x < 0.
- The second equation, f(x) = -1, is defined for 0 ≤ x ≤ 3.
- The third equation, f(x) = -2x, is defined for x > 3.
So, the domain of the function is the union of these intervals. Therefore, the domain for this function is (-∞,0) ∪ [0,3] ∪ (3,∞).
b) Finding the range:
To find the range of the given function, we need to determine the set of all possible values that the function can take. From the equation, we can see that the range of the function is [-1,∞) since -1 is the minimum value that the function takes, and there is no upper limit.
c) Finding the intercepts:
- x-intercept: To find the x-intercept, we need to determine where the graph intersects the x-axis. In this case, the x-intercept occurs when y = 0.
- For the first equation, -x+1 = 0 gives us x = 1.
- For the second equation, -1 = 0 gives us no x-intercept (since the line is constant and never intersects the x-axis).
- For the third equation, -2x = 0 gives us x = 0.
Therefore, the x-intercept is 0 and 1.
- y-intercept: To find the y-intercept, we need to determine where the graph intersects the y-axis. In this case, the y-intercept occurs when x = 0.
- For the first equation, substituting x = 0 gives us -0 + 1 = 1.
- For the second equation, substituting x = 0 gives us -1.
- For the third equation, substituting x = 0 gives us -2(0) = 0.
Therefore, the y-intercept is 1 and -1.
d) Continuity:
The function is continuous on its domain as there are no jumps, holes, or vertical asymptotes within the given intervals.
e) Graphing the function:
To graph the function, plot the points and connect them with lines. Here is a rough sketch of the graph:
```
^
2 |
| /|
1| / |
| / |
0|/ |______
|---|------|---|---
-1 0 1 3 x
```
Note that the graph consists of three parts: a line with a slope of -1 for x < 0, a horizontal line at y = -1 for 0 ≤ x ≤ 3, and a line with a slope of -2 for x > 3.