Question

The graph represents the piecewise function?

Answer 1

Well, we have a parabola that goes until x = 2, yet it does not reach that value. x = 2 starts on the line above when x = 2 and ends BEFORE x = 4. This could be modeled by:

{ f(x) = x^2, x < 2
{ f(x) = 5, 2 ≤ x < 4

Answer 2

Yes, the graph represents the piecewise function.

Explanation:

If the graph divide into two pieces that it is a piecewise function. Since the given graph is available in two ices therefore it is a piecewise function.

From the graph it is noticed that the function is parabolic before x=2 and after that the function have constant value 5 from x=2 to x=4 excluding 4.

Since the value of the function is constant for
`2\leq x<4`, so

`f(x)=5 \text{ for }2\leq x<4`

The vertex of the parabola is (0,0) and the other point on the parabola is (-2,4).

So the equation of parabola is,

`y=a(x-h)^2+k`

Where (h,k) is vertex.

`y=a(x)^2`

Since (-2,4) lies on parabola.

`4=a(-2)^2`

`a=1`

So, the equation of parabola is
`y=x^2`. Since the function is parabolic before x=2,

`f(x)=x^2\text{ for }x<2`

`f(x)=\begin{cases}x^2 & \text{ if } x<2 \\ 5 & \text{ if } 2\leq x<4\end{cases}`

Therefore the graph is a piecewise function.