Notice that each segment is the translation of the previous segment $1$ unit to the right.

Let's write the rule of the first segment and then we can write the rules of the other segments by translating it. The domain of the first segment is between

- 1

and

0 .

However,

- 1

and

0

are not included in the domain, because the endpoints are represented by open points.

\underline{Domain} - 1 < x < 0

Next, we will write a function rule. We see that the endpoints of segment are

(- 1, 0)

and

(0, 2) .

With this information we can find its slope using the Slope Formula.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(- 1, 0)$ & $(0, 2)$

m = \frac{2 - 0}{0 - ( - 1 )}

SubNeg

$a - (- b) = a + b$

m = \frac{2 - 0}{0 + 1}

AddSubTerms

Add and subtract terms

m = \frac{2}{1}

CalcQuot

Calculate quotient

m = 2

The slope of the segment is

2 .

Notice that one of the endpoints,

(0, 2),

is on the

y -

axis, which means the

y -

intercept is

2 .

In this case we can write the rule of the segment in slope-intercept form.

1 st segment f (x) = 2 x + 2

Now that we know the rule for the first segment, we can write the rule for the second one by performing a translation. The second segment is translated 1 unit to the right and that translation is equivalent to

subtracting

one

from the input.

2 nd segment 2 (x - 1) + 2 \Leftrightarrow 2 x

Similarly, we can find the rules for the remaining segments by translating the previous segment one unit to the right. To do so, we

subtract

one

from the input in the rule for the previous segment.

3 rd segment : 4 th segment : 5 th segment : 6 th segment : 2 (x - 1) - 2 \Leftrightarrow 2 x - 2 2 (x - 1) - 2 \Leftrightarrow 2 x - 4 2 (x - 1) - 4 \Leftrightarrow 2 x - 6 2 (x - 1) - 6 \Leftrightarrow 2 x - 8

Finally, let's write the piecewise function considering the domain of each piece.

f (x) = ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ 2 x + 2, for - 1 < x < 0 2 x, for 0 < x < 1 2 x - 2, for 1 < x < 2 2 x - 4, for 2 < x < 3 2 x - 6, for 3 < x < 4 2 x - 8, for 4 < x < 5

Exercise

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