Help me find the range

[thickmax]

Find the range of the function f(x)=12−4x−x2 which has the domain x∈R

Please help me understand where I am going wrong here.

The answer is f(x)⩽16 but I'm really struggling to understand how.

If I complete the square I get - (x+2)^2 +20

But when x=16, my f(x) = -304.
When x = -1 my f(x) = 19

I think I'm missing something obvious, but I cannot figure it out

 

[pka]

Find the range of the function f(x)=12−4x−x2 which has the domain x∈R
Please help me understand where I am going wrong here.
The answer is f(x)⩽16 but I'm really struggling to understand how.

You posted this in the advanced mathematics forum. So you know calculus.
Because \(f\) is a parabola find the maximum, it will be \(16\).
So the range is \(\le 16\).

 

[Dr.Peterson]

If I complete the square I get - (x+2)^2 +20

Please show how you got that. If you expand it, you get -x^2 - 4x + 16, which is not what you want. So your error is in that work.

 

[thickmax]

Please show how you got that. If you expand it, you get -x^2 - 4x + 16, which is not what you want. So your error is in that work.

-X^2-4X+12
-(X^2+4X)+12
-((X+2)^2-8)+12
-(X+2)^2+20

Is how I've worked it out....


But I still don't understand how X≤16

Even if I used the -X^2-4x+16 as the function,

I get

X Y
-1 = 11
0 = 12
1 = 11
2 = 8
3 = 1

Please help

 

[thickmax]

You posted this in the advanced mathematics forum. So you know calculus.
Because \(f\) is a parabola find the maximum, it will be \(16\).
So the range is \(\le 16\).

Thank you for the clarification.

but how is X≤16 known?

I'm getting major variations above and below 16 when I run the function, so I do not understand how it is know that X≤16

 

[Dr.Peterson]

-X^2-4X+12
-(X^2+4X)+12
-((X+2)^2-8)+12
-(X+2)^2+20

But X^2+4X is not equal to (X+2)^2-8 ! Check every step!

But I still don't understand how X≤16

Even if I used the -X^2-4x+16 as the function,

No, don't change the problem; change your work!!

The correct result is -(X+2)^2+16. Do you see that this is never greater than 16?

I'm getting major variations above and below 16 when I run the function, so I do not understand how it is know that X≤16

For what value of x do you get a result greater than 16 (using the function as given to you)?

 

[thickmax]

But X^2+4X is not equal to (X+2)^2-8 ! Check every step!

No, don't change the problem; change your work!!

The correct result is -(X+2)^2+16. Do you see that this is never greater than 16?

For what value of x do you get a result greater than 16 (using the function as given to you)?

Thank you for the explanation,

I get the answer now-(X+2)^2 will always equal zero or a negative number. So the answer will never be over 16.

But I'm still not sure how completing my square went wrong,

But the 8 should be a 4.

My example had an was

-X^2-8X+7
-(X^2+8X)+7
-((X+4)^2-16)+7
-(X+4)^2+16+7
-(X+4)^2 +23

I though when you go from step 2 to 3, the 8 (or in the original question 4) was doubled, but it's meant to be the number in the inner bracket, multiplied by the square root.

Understood. I will practice my expanding the square more I think!

Many thanks for the help Dr Peterson

 

[Dr.Peterson]

My example had an was

-X^2-8X+7
-(X^2+8X)+7
-((X+4)^2-16)+7
-(X+4)^2+16+7
-(X+4)^2 +23

I though when you go from step 2 to 3, the 8 (or in the original question 4) was doubled, but it's meant to be the number in the inner bracket, multiplied by the square root.

The 16 they subtract is the SQUARE of the 4; it is not TWICE the 8! [Your "the number in the inner bracket, multiplied by the square root" doesn't quite make sense.]

You have to understand the meaning of the process, which is to undo the expansion of the square. Observe that (x+4)^2 = x^2 + 8x + 16, where the middle term comes from doubling the 4 (because the middle term of (a+b)^2 is 2ab), and the last term comes from squaring the 4. So going in reverse, we see x^2 + 8x and divide the 8 by 2 to get the 4, and write (x+4)^2. But this would have an extra term of 16, so we have to subtract that to avoid changing the meaning of the expression. That is, in replacing x^2 + 8x with (x+4)^2, we have added 16 to the expression, so we compensate by subtracting 16.