Question

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

a. [3, 00)
b. [2,00)
c. [3,6]
d. (-00,00)

(00 is infinite

Answer 1

correct answer is B

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Answer 2

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.