Question

Please help with this question

Answer 1

Answer: (2) =

(−3) =

(−1) =

(−4) =

(0) =

(−4) =

(−1) =

(3)  =

() = �

−2 + 1, < 0

2

3

− 3, ≥ 0 () = �

5, ≤ 2

2 − 4, >

Step-by-step explanation: Piecewise Function

(a solid dot means "including",

an open dot means "not including")

And this is how we write it:

Piecewise Function

The Domain (all the values that can go into the function) is all Real Numbers up to and including 6, which we can write like this:

Dom(f) = (-∞, 6] (using Interval Notation)

Dom(f) = {x member of Reals | x ≤ 6} (using Set Builder Notation)

And here are some example values:

X Y

−4 16

−2 4

0 0

1 1

2 6

3 7

Example: Here is another piecewise function:

continuous jump fn h(x) = 2 if x<=1, x if x>1

which looks like:   continuous jump graph h(x)

What is h(−1)?

x is ≤ 1, so we use h(x) = 2, so h(−1) = 2

What is h(1)?

x is ≤ 1, so we use h(x) = 2, so h(1) = 2

What is h(4)?

x is > 1, so we use h(x) = x, so h(4) = 4

Piecewise functions let us make functions that do anything we want!

Example: A Doctor's fee is based on the length of time.

Up to 6 minutes costs $50

Over 6 and up to 15 minutes costs $80

Over 15 minutes costs $80 plus $5 per minute above 15 minutes

Which we can write like this:

function piecewise f(t)=50 if t<=6, 80 if t>6 and t<=15, 80 + 5(t-15) if t>15

You visit for 12 minutes, what is the fee? $80

You visit for 20 minutes, what is the fee? $80+$5(20-15) = $105

The Absolute Value Function

The Absolute Value Function is a famous Piecewise Function.

It has two pieces:

below zero: -x

from 0 onwards: x

Absolute Value function

f(x) = |x|

Absolute Value function

The Floor Function

The Floor Function is a very special piecewise function. It has an infinite number of pieces:

Floor function

The Floor Function