Question

The graph represents the piecewise function

Look at the picture

There are three answers in total

Answer 1

Piecewise function can be explained as a set of subfunctions (each subfunction applied to an interval). Here we can observe a piecewise function that is composed of three sub functions.

-The first subfunction: between 0 and 3 (open interval in 3) there is a quadratic function, we can obtain it from the general form of the quadratic function when it has a vertex on (0,0) which is:

`f(x)=ax^(2)`

`a` will be a positive value if the parabola is opening upward and negative if it is opneing downward. This one opens upward so it is positive and it will be =1 because when we make
`x=1` on the graph corresponds to
`y=1` so:

`f(x)=ax^(2)\\1=a(1^(2))\\a=1`

Our subfunction if
`0\leq x<3` is
`f(x)=x^(2)`

-the second and third subfunctions: both are straight lines defined each in an interval. The first between 3 and 6 (open interval in 3) and the second between 6 and 10. To build this two functions we need to understand the general form of the line function:

`f(x)=mx+b`

where m is the slope of the line and b is the y intercept.

we calculate first m and b for both lines with the m and b equation (we must take two coordintates of each line (
`x_(1),y_(1),x_(2),y_(2)`):

`m=(y_(2)-y_(1))/(x_(2)-x_(1))`

`b=y-mx`

First line (4,6),(5,5)

`m=(5-6)/(5-4)}=-1`

`b=(5)-(-1)(5)=10`

Second line (8,7),(10,10)

`m=(10-7)/(10-8)}=(3)/(2)}=1.5`

`b=(10)-((3)/(2)})(10)=-5`

Now we can build both the equations:

First Lline (
`3<x\leq 6`)

`f(x)=mx+b\\f(x)=(-1)x+10\\f(x)=10-x`

Second Line(
`6\leq x\leq 10`):

`f(x)=mx+b\\f(x)=((3)/(2)})x-5`

Answer 2

Drop-down menu selections are not provided.

f(x) = {x^2, 0 ≤ x < 3
.. .. .. {10 -x, 3 < x ≤ 6
.. .. .. {(3/2)x -5, 6 ≤ x ≤ 10

___
Note that the last two function segments both give (6, 4), but it is bad form to define a piecewise function such that the pieces overlap. That is, f(6) has two definitions.