WRITE A PIECEWISE FUNCTION FROM A GRAPH

Write a piecewise function for the graph.

Problem 1 :

Solution :

Equation of horizontal line:

y = 2

x ≥ 0

Equation of raising line:

Slope = 1/1 = 1

y = mx + b

y = x + b --(1)

The raising line is passing through the point (-2, 0).

0 = -2 + b

b = 2

Applying the value of b in (1), we get

y = x + 2

So, the required piecewise function is

Problem 2 :

Solution :

Equation of horizontal line:

y = -3

x ≤ 0

Equation of raising line:

Slope = 3/1 = 3

y = mx + b

y = 3 x + b --(1)

The raising line is passing through the point (1, 0).

0 = 3(1) + b

b = -3

Applying the value of b in (1), we get

y = 3x - 3 for x > 0

So, the required piecewise function is

Problem 3 :

Solution :

Both pieces are falling lines.

Equation of first falling line :

Slope = -1/1 ==> -1, y-intercept = 0

y = -x + 0

y = -x

x < 4

Equation of second falling line:

Slope = -1/1 ==> -1

y = -x + b ---(1)

The line is passing through the point (5, -4)

By applying (5, -4) in (1), we get

-4 = -5 + b

-4 + 5 = b 

b = 1

Applying the value of b in (1), we get

y = -x + 1

So, the required piecewise function is

Problem 4 :

Solution :

Both pieces are raising lines.

Equation of first raising line :

Slope = 2/1 ==> 2

y = 2x + b -----(1)

This line is passing through the point (-3, -4)

Applying the point (-3, -4) in (1), we get

-4 = 2(-3) + b

b = -4 + 6

b = 2

By applying b = 2, we get

y = 2x + 2 for x ≤ -2

Equation of second raising line:

Slope = 1/2

y = (1/2)x + b ---(2)

The line is passing through the point (0, -1)

By applying (0, -1) in (2), we get

-1 = (1/2)(0) + b

-1 = b

Applying the value of b in (2), we get

y = (1/2)x - 1 for x > -2

So, the required piecewise function is

Problem 5 :

Solution :

There are three pieces.

Equation of horizontal line :

y = 1 for x ≤ -2

Equation of raising line :

Slope = 2/1 ==> 2

y = 2x + b

This line is passing through the point (-1, -2)

Applying the point (-1, -2) in (1), we get

-2 = 2(-1) + b

b = -2 + 2

b = 0

By applying b = 2, we get

y = 2x for -2 < x ≤ 0

Equation of falling line:

Slope = -1/2

y = (-1/2)x + b ---(2)

The line is passing through the point (2, 1)

By applying (2, 1) in (2), we get

1 = (-1/2)(2) + b

1 = -1 + b

b = 2

Applying the value of b in (2), we get

y = (-1/2)x + 2 for x > 0

So, the required piecewise function is

Problem 6 :

Solution :

There are three pieces.

Equation of horizontal line :

y = 3 for x ≥ 3

Equation of raising line :

Slope = 1/1 ==> 1

y = x + b

This line is passing through the point (-2, 2)

Applying the point (-2, 2) in (1), we get

2 = -2 + b

b = 4

By applying b = 4, we get

y = x + 4 for -2 < x ≤ -1

Equation of falling line:

Slope = -1/4

y = (-1/4)x + b ---(2)

The line is passing through the point (3, -1)

By applying (3, -1) in (2), we get

-1 = (-1/4)(3) + b

-1 = -3/4 + b

b = -1/4

Applying the value of b in (2), we get

y = (-1/4)x + (-1/4) for -1 < x < 3

So, the required piecewise function is

Problem 7 :

Solution :

From the graph, it is a step function.

y = -1 for -1 ≤ x < 1

y = -3 for -3 ≤ x < -1

y = -5 for -5 ≤ x < -3

So, the required piecewise function is

Problem 8 :

Solution :

From the graph, it is a step function.

y = 4 for 0 < x ≤ 1

y = 3 for 1 < x ≤ 2

y = 2 for 2 < x ≤ 3

y = 1 for 3 < x ≤ 4

So, the required piecewise function is