Answer:
f(x) = {1/2x +3 1/2, for x < 1; -2x +6, for x ≥ 1}
intercepts: (-7, 0), (3, 0), (0, 3 1/2)
Step-by-step explanation:
function definition
The point-slope equation of a line is ...
y -k = m(x -h) . . . . line with slope m through point (h, k)
When we solve this for y, we get ...
y = m(x -h) +k
The slope of a line is the ratio of rise to run. For the left side of the function, the slope is ...
m = 1/2
For the right side of the function, the slope is ...
m = -2/1 = -2
The point (1, 4) is common to both parts of the function, so we can write the piecewise function as ...
[tex]f(x)= \begin{cases}\dfrac{1}{2}(x-1)+4,&\text{ for $x<1$}\\-2(x-1)+4,&\text{ for $x\ge1$}\end{cases}[/tex]
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intercepts
Two of the intercepts are shown on the graph. The y-intercept is in the left side of the function, so will be f(0) = 1/2(-1) +4 = 3 1/2.
The x-intercept of the left side of the function is not shown on the graph. We can find it by determining x for f(x) = 0.
f(x) = 0 = 1/2(x -1) +4
-4 = 1/2(x -1) . . . add 4
-8 = x -1 . . . . . multiply by 2
-7 = x . . . . . . add 1
The x-intercepts are (-7, 0) and (3, 0).
The y-intercept is (0, 3.5).
