Given:
A piecewise function
[tex]f(x)=\begin{cases}3 &,x<0 \\ x^2+2 &,0\leq x<2 \\ \dfrac{1}{2}x+5 &,x\geq 2 \end{cases}[/tex]
To find:
The range of the function.
Solution:
Range is the set of output values.
For [tex]x<0[/tex], the function is [tex]f(x)=3[/tex] ...(i)
For [tex]0\leq x<2[/tex], the function is [tex]f(x)=x^2+2[/tex].
[tex]0\leq x<2[/tex]
Squaring each side.
[tex]0^2\leq x^2<2^2[/tex]
Adding 2 on each side.
[tex]0+2\leq x^2+2<4+2[/tex]
[tex]2\leq f(x)<6[/tex] ...(ii)
For [tex]x\geq 2[/tex], the function is [tex]f(x)=\dfrac{1}{2}x+5[/tex].
[tex]x\geq 2[/tex]
Divide both sides by 2.
[tex]\dfrac{1}{2}x\geq 1[/tex]
Adding 5 on each side.
[tex]\dfrac{1}{2}x+5\geq 1+5[/tex]
[tex]f(x)\geq 6[/tex] ...(iii)
From (i), (ii) and (iii), it is clear that the values of f(x) lies in the interval [2,∞).
So, the range of the given function is [2,∞).
Therefore, the correct option is b.
