Answer:
The correct option is 3.
Step-by-step explanation:
The given piecewise function is
[tex]f(x)=\begin{cases}3x^2+1 & \text{ if } -4<x<6 \\ 6 & \text{ if } 6\leq x<9 \end{cases}[/tex]
Range is the set of output or y values.
The given function for 6 ≤ x < 9 is
[tex]f(x)=6[/tex]
It is a constant function, the value of function is 6 for all values of x.
Range = 6
The given function for -4 < x < 6 is
[tex]f(x)=3x^2+1[/tex] .... (1)
It is a quadratic function.
The vertex form of a quadratic function is
[tex]f(x)=a(x-h)^2+k[/tex] ....(2)
Where (h,k) is vertex and a is constant.
From (1) and (2), we get a=3,h=0,k=1.
The vertex of this function is (0,1), it means the range of this function is greater than or equal to 1. But this function is only defined for -4 < x < 6.
[tex]f(-4)=3(-4)^2+1=49[/tex]
[tex]f(6)=3(6)^2+1=109[/tex]
The maximum value of the maximum value of the function is 109 at x=6. Since 6 is not included in the interval -4 < x < 6, therefore 109 is not included in the range.
Range = [1,109)
When we combined the range of both functions we get
Range = [1,109)
Therefore the correct option is 3.
