How do you find the range of 1/{(x-1)(x+2)} for [0, 6]?

[solve]

Homework Statement

Need to find the range of 1/{(x-1)(x+2)} for [0, 6]

Homework Equations

The Attempt at a Solution

The domain is [0, 1) AND (1, 6] because 1/0 is an invalid operation. So I tried and this is what I got: [-0.5, ∞) and [0.025, ∞). Wrong, of course :)

How do I go about arriving to the right solution?

Thanks.

 

[SammyS]

Homework Statement

Need to find the range of 1/{(x-1)(x+2)} for [0, 6]

Homework Equations

The Attempt at a Solution

The domain is [0, 1) AND (1, 6] because 1/0 is an invalid operation. So I tried and this is what I got: [-0.5, ∞) and [0.025, ∞). Wrong, of course :)

How do I go about arriving to the right solution?

Thanks.

Can you find the range of (x-1)(x+2) for [0, 1) ∪ (1, 6] ?

 

[solve]

Can you find the range of (x-1)(x+2) for [0, 1) ∪ (1, 6] ?

Hopefully I am right: [-2,0) and (0,40]

 

[SammyS]

Hopefully I am right: [-2,0) and (0,40]

Yes, that's correct.

Do you see how that's related to the range of 1/((x-1)(x+2)) ?

 

[solve]

Yes, that's correct.

Do you see how that's related to the range of 1/((x-1)(x+2)) ?

Honestly not. Can I get a hint or something :)

 

[Mentallic]

Honestly not. Can I get a hint or something :)

If the values of a variable k range between 2 to 5, then what are the possible values of 1/k?

 

[solve]

If the values of a variable k range between 2 to 5, then what are the possible values of 1/k?

[-0.5, infinity) and [0.025, infinity) ?

But it's supposed to be (-infinity, -0.5] instead of [-0.5, infinity) so I was wondering how 1/0 gives negative infinity. This is what originally confused me.

 

[Mentallic]

If the values of a variable k range between 2 to 5, then what are the possible values of 1/k?

[-0.5, infinity) and [0.025, infinity) ?

Well no but I see we're just going to disregard my example question since you already have the hang of it.

But it's supposed to be (-infinity, -0.5] instead of [-0.5, infinity) so I was wondering how 1/0 gives negative infinity. This is what originally confused me.

Well for the quadratic (x-1)(x+2) with $x\in [0,1)$, the range is negative. So for the reciprocal function 1/((x-1)(x+2)) it's still going to be negative, thus this is how you should quickly realize that as x approaches 1 from the left side (0.99... etc.), it should be approaching $-\infty$ rather than $+\infty$

 

[solve]

Well for the quadratic (x-1)(x+2) with $x\in [0,1)$, the range is negative. So for the reciprocal function 1/((x-1)(x+2)) it's still going to be negative, thus this is how you should quickly realize that as x approaches 1 from the left side (0.99... etc.), it should be approaching $-\infty$ rather than $+\infty$

Whoa, this is WAY above my head. $x\in [0,1)$ for one. Have no idea what that's supposed to mean. Can you, please, tell me what that is? Thank You.

 

[Mentallic]

Whoa, this is WAY above my head. $x\in [0,1)$ for one. Have no idea what that's supposed to mean. Can you, please, tell me what that is? Thank You.

It's nothing special, it just means the values of x in that set (0 to 1, but not including 1). It means the same thing as what Sammy posted in post #2.

 

[solve]

It's nothing special, it just means the values of x in that set (0 to 1, but not including 1). It means the same thing as what Sammy posted in post #2.

Thank You very much, Mentallic. Will look into it a bit closer.

 

[SammyS]

[-0.5, infinity) and [0.025, infinity) ?

But it's supposed to be (-infinity, -0.5] instead of [-0.5, infinity) so I was wondering how 1/0 gives negative infinity. This is what originally confused me.

For number, z, where z is between -2 and 0, what is 1/z? For one thing, isn't 1/z negative if z is negative.

... And if z is a negative number very close to zero, such as z=-0.001, isn't 1/z a very "large" negative number, such as -1000 ?

 

[solve]

For number, z, where z is between -2 and 0, what is 1/z? For one thing, isn't 1/z negative if z is negative.

... And if z is a negative number very close to zero, such as z=-0.001, isn't 1/z a very "large" negative number, such as -1000 ?

Oh, now I understand the flaw in my thinking- I kept imagining a positive 1 between -2 and 0, for some reason. Thank You, SammyS.