2 Answers

Billy Kimble · Answer 1 · 2022-04-30T22:12:10+0000

Given:

A piecewise function

$f(x)=\begin{cases}3 &,x<0 \\ x^2+2 &,0\leq x<2 \\ (1)/(2)x+5 &,x\geq 2 \end{cases}$

To find:

The range of the function.

Solution:

Range is the set of output values.

For
x<0 , the function is
f(x)=3 ...(i)

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

$0\leq x<2$

Squaring each side.

$0^2\leq x^2<2^2$

Adding 2 on each side.

$0+2\leq x^2+2<4+2$

$2\leq f(x)<6$ ...(ii)

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

$x\geq 2$

Divide both sides by 2.

$(1)/(2)x\geq 1$

Adding 5 on each side.

$(1)/(2)x+5\geq 1+5$

$f(x)\geq 6$ ...(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.

Please log in or register to add a comment.