ASAP Help with calculus (finding domain of a natural log) homework question

[avie_j]

Equation: f(x)= ln(4-sqrt(x-4))

ANS:
a) (4, 20] b) (-infinity,20) c) (-infinity,20] d) [4,infinity) e) [4,20) f) (6,22] g) (4,infinity) h) [6,22)

Attempt:

4-sqrt(x-4) > 0 ...was told to put the part after the natural log to greater than 0
4 > sqrt(x-4)
(4)^2 > (sqrt(x-4))^2 ... take away the sqrt
16 > x-4
16 + 4 > x
20 > x

(( I got that x is greater than 4, but do not know why to do this ))

- sqrt(x-4) > 0 ... divide by -1
sqrt(x-4) > 0 ...to the power of 2
x-4 > 0
x > 4

So I know the function is less than 20, but according to it's graph, its greater than 4.

I do not know how to find if the 20 or 4 is what x is also equal to... I have a feeling it's the 4

The only advice I got was that the "thing inside the square root can't be negative"

I know you can't square root a negative but I'm confused as what my teacher means by that.

ANY guidance is appreciated! Thank you

 

[Otis]

I got that x is greater than 4, but do not know why to do this

Hi Avie. What does "this" mean? Finding the domain of sqrt(x-4) or the exercise itself (i.e., finding the domain of function f).

Some values of x won't work as inputs to certain functions because they cause the function to produce undefined or Imaginary outputs. Finding domains is about eliminating x-values that cause problems.

By the way, "x is greater than 4" is not quite correct because sqrt(4-4) does exist -- that is, sqrt(4-4) is a Real output.

So I know the function is less than 20, but according to it's graph, its greater than 4. I do not know how to find if the 20 or 4 is what x is also equal to... I have a feeling it's the 4

Maybe you're thinking correctly, but your first statement above is not true. We're talking about values in the domain, not function values. We're also not solving an equation for x or finding "what x is equal to").

Before we go any further, can you state in your own words what a function domain is? I'd like to confirm that we're on the same page. Thanks.

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[avie_j]

Hi! Thanks for the reply. I'll try to explain the best I can.

I think of a function's domain as all the possible x-values of the function.

So something similar to 4 < x < 20 would be the domain... at least that's how it was explained to me

 

[Otis]

I think of a function's domain as all the possible x-values of the function.

Very good. The domain here is a set of input numbers for which the output number f(x) is Real.

The function in this exercise is a composite function (a function inside a function). The natural log function's input itself contains a square root function. Because of Order of Operations, we want to make sure that sqrt(x-4) is defined first. That is, we work from innermost function to outermost function.

Have I helped you to understand why we do "this"?

something similar to 4 < x < 20 would be the domain

Yes, that is very close to correct.

You know that -- in the Real number system -- we cannot take the square root of a negative number. But we can take the square root of zero.

In other words, x-4 must be zero or positive (zero or positive is what "not negative" means), so you ought to have used the greater-than-or-equal-to inequality sign when setting up your x-4 inequality to solve.

Do you have any more questions?

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[Otis]

- sqrt(x-4) > 0 ... divide by -1
sqrt(x-4) > 0 ...to the power of 2
x-4 > 0

I forgot to mention something.

Don't write the subtraction operator in front of sqrt(x-4). That is, it's not a negative sign, so you don't need to divide or multiply both sides by -1.

(If you ever do need to multiply or divide any inequality by -1, don't forget the very important rule: Change the direction of the inequality symbol.)

Just fix the inequality symbol here, before solving for x:

sqrt(x-4) > 0

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[avie_j]

Yes, you have helped me understand "this". Thank you very much, you can explain concepts much better!

--- Just saw that you answered this---
Just to make sure, when I set sqrt(x-4) to "greater-than-or-equal-to 0", it would be:
sqrt(x-4) greater-than-or-equal-to 0 AND NOT
-sqrt(x-4) greater-than-or-equal-to 0 ?

Also, if you don't mind, I would like to understand when to use > or "greater-than-or-equal-to"
better. From my current understanding:

If a function must be zero or positive, use "greater-than-or-equal-to"
If a function is positive: greater-than
If a function is negative: less-than
If a function must be zero or negative, use "less-than-or-equal-to" (?)

Like what if the question was sqrt(x+4) instead? I would also use "greater-than-or-equal-to" ?

 

[Otis]

Just to make sure, when I set sqrt(x-4) to "greater-than-or-equal-to 0", it would be:
sqrt(x-4) greater-than-or-equal-to 0 AND NOT
-sqrt(x-4) greater-than-or-equal-to 0 ?

Yes, we want to be sure that sqrt(x-4) represents a Real number, before it gets subtracted from 4. That subtraction is not part of the square-root function.

We can think like this:

ln(4 - c) where c represents the number sqrt(x-4). We want c to be a Real number, right? We don't care about the subtraction until we work on the domain of the outer function ln().

If a function input must be zero or positive, use "greater-than-or-equal-to"
If a function input is positive: greater-than
If a function input is negative: less-than
If a function input must be zero or negative, use "less-than-or-equal-to" (?)

You're not using the right vocabulary. When we say something like, "The function must be positive" we mean f(x), not x. In other words, if you say "function", then you're talking about the outputs.

The domain is a set of inputs. Those are x-values, not function values. Therefore we need to say "function inputs" above, as I've shown in red.

If the input for some function must be zero or negative, then we would write less-than-or-equal-to 0.

what if the question was sqrt(x+4) instead? I would also use "greater-than-or-equal-to"?

Yes. The input x+4 must be non-negative, so we would write the inequality as x+4 greater than or equal to 0.

sqrt(anything) means the expression 'anything' must not be negative -- when working within the Real-number system.

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[avie_j]

Okay, thank you so much!
I've realized my knowledge is a bit behind due to online learning and keep trying to catch up. You are amazing!

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[Otis]

4-sqrt(x-4) > 0

You did good solving that one.

ln() functions require their inputs to be positive (for Real outputs), so you've used the correct inequality symbol.

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[Otis]

I've realized my knowledge is a bit behind due to online learning and keep trying to catch up.

You're welcome. Keep practicing. Do extra problems, whenever you have time. (Hopefully your text has an answer section.) Repeated exposure to the same patterns is how we grow our brain.

See you next time.

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