Understanding Negative Numbers in -2x |X| < 4 Equation

[brycenrg]

Homework Statement

This is the solution, the question was find its domain.

Homework Equations

How does |X| (less than or equal to) 4, when a negative number is inputed into -2x how does that = a positive number?

The Attempt at a Solution

On the graph to me All X values < 0 should be negative or atleast until -5
Because the -2x if |x| < 4

If we put f(2) -2 * |2| = -4 which works on the graph but if I put -2 which would equal positive 2 because of the absolute value the graph seems to not make sense for me.
f(-2) should equal -4 as well right?

 

[Mark44]

Homework Statement

View attachment 67376
This is the solution, the question was find its domain.

Homework Equations

How does |X| (less than or equal to) 4, when a negative number is inputed into -2x how does that = a positive number?

f(-2) = -2(-2) = + 4
f(3) = -2(3) = -6

The Attempt at a Solution

On the graph to me All X values < 0 should be negative or atleast until -5

I think that you are confusing values in the domain with the resulting function values.

The function has three different formulas, with each valid on a different interval. For input values between -4 and 4, the middle formula is used.

Because the -2x if |x| < 4

If we put f(2) -2 * |2| = -4 which works on the graph but if I put -2 which would equal positive 2 because of the absolute value the graph seems to not make sense for me.
f(-2) should equal -4 as well right?

You are not graphing y = |x|, which seems to be part of your confusion here. They could just as well have said that the middle formula applies if -4 ≤ x ≤ 4.

 

[brycenrg]

Thanks for the reply, but aren't you graphing -2*|x| for the interval -4 ≤ x ≤ 4?
For example f(-3) = -6 because -2 *|-3| = -6?
thats why I am thinking the graph at -4 ≤ x ≤ 0 should be negative numbers not positive

 

[Dick]

Thanks for the reply, but aren't you graphing -2*|x| for the interval -4 ≤ x ≤ 4?
For example f(-3) = -6 because -2 *|-3| = -6?
thats why I am thinking the graph at -4 ≤ x ≤ 0 should be negative numbers not positive

No. The problem says for |x|<=4, f(x)=-2x. That's not the same as saying for |x|<=4, f(x)=-2|x|.

 

[brycenrg]

I see I am confused. I thought the right side |x| <= 4 means any value less than or equal to 4 but because its x is in absolute value brackets if x was -1 it would be 1 so then you would plug 1 into -2x? What is the point of the |x| then? Does that make sense on my confusion lol?

 

[Mark44]

I see I am confused. I thought the right side |x| <= 4 means any value less than or equal to 4

That's not what |x| ≤ 4 means. -5 ≤ 4, but -5 doesn't satisfy |x| ≤ 4.

but because its x is in absolute value brackets if x was -1 it would be 1 so then you would plug 1 into -2x?

NO.

What is the point of the |x| then?

The ONLY purpose of the absolute value here is to define the interval on which the second formula should be applied.

The second part of the function's definition could have been written as
f(x) = -2x, if -4 ≤ x -4

To answer your question above, f(-1) = -2(-1) = +2.

Does that make sense on my confusion lol?

 

[Steven60]

Always think! What numbers can I put in for x so |x| is less than 4. I like to think and teach that the absolute of any number is that same exact number but without a sign. So in |x|<4 we are asking which number or numbers, if any, when we ignore the sign will be less than 4. I will give you a hint, there are 7 integers in this set.