Question

Which of the following accurately shows the range of the function defined below? ​

Answer 1

Given:

A piecewise function

$f(x)=\begin{cases}3 &,x$

To find:

The range of the function.

Solution:

Range is the set of output values.

For $x$, the function is $f(x)=3$       ...(i)

For $0\leq x$, the function is $f(x)=x^2+2$.

$0\leq x$

Squaring each side.

$0^2\leq x^2$

Adding 2 on each side.

$0+2\leq x^2+2$

$2\leq f(x)$               ...(ii)

For $x\geq 2$, the function is $f(x)=\dfrac{1}{2}x+5$.

$x\geq 2$

Divide both sides by 2.

$\dfrac{1}{2}x\geq 1$

Adding 5 on each side.

$\dfrac{1}{2}x+5\geq 1+5$

$f(x)\geq 6$                 ...(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.