The piece-wise linear functions can be written as follows:
- [tex]f(x) = -0.5x + 3, x < 5[/tex].
- [tex]f(x) = -5, x = 5[/tex]
- [tex]f(x) = -x + 9, x > 5[/tex].
What is a linear function?
A linear function is modeled by:
y = mx + b
In which:
- m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.
- b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.
For x less than 5, the y-intercept is of b = 3 and the function also gos through (2,2), hence the slope is:
m = (2 - 3)/(2 - 0) = -0.5.
Hence the rule is:
[tex]f(x) = -0.5x + 3, x < 5[/tex].
When x = 5, the function is constant at -5, hence:
[tex]f(x) = -5, x = 5[/tex]
For x less than 5, the function goes through points (6,3) and (7,2), hence the slope is:
m = (2 - 3)/(7 - 6) = -1.
Then:
y = -x + b
When x = 6, y = 3, hence the y-intercept is given as follows:
3 = -6 + b
b = 9.
Hence the rule is:
[tex]f(x) = -x + 9, x > 5[/tex].
More can be learned about linear functions at https://brainly.com/question/25537936
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