Question

The range of which function includes -4?


`y= \sqrt{x` -5

`y= \sqrt{x` +5

`y= \sqrt{x +5`

`y= \sqrt{x -5`

Answer 1

The correct option is 1.

Explanation:

We know that the range of a radical function is greater than 0.

`√(x)>0` .... (1)

`√(x+5)>0`

`√(x-5)>0`

The range of third and fourth function is greater than 0, therefore the range does not includes -4. So, option 3 and 4 are incorrect.

Subtract 5 from both sides.

`√(x)-5>-5`

`y>-5`

It means the range of the fist function is greater than -5, therefore the range includes -4. So, option 1 is correct.

Add 5 on both sides.

`√(x)+5>5`

`y>5`

It means the range of the fist function is greater than 5, therefore the range does not includes -4. So, option 2 is incorrect.

Answer 2

The range of a function is the set of all values (outputs) assumed by the dependent variable. Thus, according to this statement we can affirm that the correct answer of this question is the equation given by:


`y=√(x)-5`

So:


`If \ y=-4 \\ \\ Then \ -4=√(x)-5 \\ \\ \therefore √(x)=1 \\ \\ \therefore \boxed{x=1}`

Let's find the range of the other functions to contrast this conclusion:


`y=√(x)+5\to R:y\geq5\\\\y=√(x+5)\to R:y\geq0\\\\y=√(x-5)\to R:y\geq0`

So as shown in the figure below the range of this function includes -4