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**Q1.**I was just wondering that since and are just reflections of each other in the x-axis, then can I make the statement:

Since and are just reflection of each other in the x-axis, their range will be the same provided their domain is the same.

So, e.g. if we take the domain as the real numbers, since is greater than 0 for all x in the domain, then by the statement, must also be greater than 0 for all x in the domain.

**Q2.**If I have a function f(x) and I claim it's gradient is positive throughout its domain i.e. it is an increasing function. Let's say the gradient is all of the real numbers again. Then, if I claim it's second derivative f''(x) is negative when x is negative. Does this mean that in the range (- inf, 0), when x decreases, its gradient increases and when x increases, its gradient decreases?

Last edited by Takeover Season; 11 months ago

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Since and are just reflection of each other in the x-axis, their range will be the same provided their domain is the same.

So, e.g. if we take the domain as the real numbers, since is greater than 0 for all x in the domain, then by the statement, must also be greater than 0 for all x in the domain.

**Takeover Season**)**Q1.**I was just wondering that since and are just reflections of each other in the x-axis, then can I make the statement:Since and are just reflection of each other in the x-axis, their range will be the same provided their domain is the same.

So, e.g. if we take the domain as the real numbers, since is greater than 0 for all x in the domain, then by the statement, must also be greater than 0 for all x in the domain.

True, if then both functions have the same range.

But if say , then but . These ranges are not the same just because their domain is the same!

A more accurate statement would be that and have the same range provided their domains are reflections about .

I.e. if then . If I take the reflection about the the domain is then and hence as well.

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Q2.

If I have a function f(x) and I claim it's gradient is positive throughout its domain i.e. it is an increasing function. Let's say the gradient is all of the real numbers again. Then, if I claim it's second derivative f''(x) is negative when x is negative. Does this mean that in the range (- inf, 0), when x decreases, its gradient increases and when x increases, its gradient decreases?

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If I'm to get my toothpick out and examine your statement with precise accuracy, then you are not technically correct.

True, if then both functions have the same range.

But if say , then but . These ranges are not the same just because their domain is the same!

A more accurate statement would be that and have the same range provided their domains are reflections about .

I.e. if then . If I take the reflection about the the domain is then and hence as well.

[b]

I don't really follow you here. At first you say f(x) is such that f'(x) > 0 but then you go on to suppose the gradient is over instead.

**RDKGames**)If I'm to get my toothpick out and examine your statement with precise accuracy, then you are not technically correct.

True, if then both functions have the same range.

But if say , then but . These ranges are not the same just because their domain is the same!

A more accurate statement would be that and have the same range provided their domains are reflections about .

I.e. if then . If I take the reflection about the the domain is then and hence as well.

[b]

I don't really follow you here. At first you say f(x) is such that f'(x) > 0 but then you go on to suppose the gradient is over instead.

f(x) and f(-x) are reflections of each other in the x-axis and therefore, provided the domains on which f(x) and f(-x) are defined are reflections about x=0, then both functions f(x) and f(-x) will have the same range.

Sorry, I meant that let's say I have a function and the domain is all of the real numbers. Then, it is an increasing function since its gradient is positive over the entire domain as for all x in the domain. This next part doesn't apply to , but just in general, let's say f(x) was an increasing function throughout its domain of the real numbers.

Then, if I claim its second derivative f''(x) is negative for x < 0 and positive for x > 0. Then, does this mean that in the interval of the domain (- inf, 0), when x decreases, its gradient increases and when x increases, its gradient decreases?

In general, I mean that let's say f''(x) is negative in the interval [a,b]. Then, if x = a and it increases to e.g. x = c which is between x = a and x = b, then I expect the gradient to lower at x = c than it was at x = a?

i.e. as x increases along the interval, the gradient will fall? ... and vice versa... if x decreases from x = b to x = a, the gradient will increase?

Last edited by Takeover Season; 11 months ago

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Thank you for the explanation, that was great! So, can I say that in general though? e.g.

f(x) and f(-x) are reflections of each other in the x-axis and therefore, provided the domains on which f(x) and f(-x) are defined are reflections about x=0, then both functions f(x) and f(-x) will have the same range.

**Takeover Season**)Thank you for the explanation, that was great! So, can I say that in general though? e.g.

f(x) and f(-x) are reflections of each other in the x-axis and therefore, provided the domains on which f(x) and f(-x) are defined are reflections about x=0, then both functions f(x) and f(-x) will have the same range.

Sorry, I meant that let's say I have a function and the domain is all of the real numbers. Then, it is an increasing function since its gradient is positive over the entire domain as for all x in the domain. This next part doesn't apply to , but just in general, let's say f(x) was an increasing function throughout its domain of the real numbers.

Then, if I claim its second derivative f''(x) is negative for x < 0 and positive for x > 0. Then, does this mean that in the interval of the domain (- inf, 0), when x decreases, its gradient increases and when x increases, its gradient decreases?

Then, if I claim its second derivative f''(x) is negative for x < 0 and positive for x > 0. Then, does this mean that in the interval of the domain (- inf, 0), when x decreases, its gradient increases and when x increases, its gradient decreases?

Clearly, if then the gradient is increasing (in the direction of x increasing, hence decreasing in the direction of x decreasing).

And if then the gradient is decreasing (in the dir. of x increasing, hence increasing in the direction of x decreasing).

Therefore, in the domain where we have , then the gradient is indeed the gradient is increasing as x decreases, and vice versa.

In general, I mean that let's say f''(x) is negative in the interval [a,b]. Then, if x = a and it increases to e.g. x = c which is between x = a and x = b, then I expect the gradient to lower at x = c than it was at x = a?

i.e. as x increases along the interval, the gradient will fall? ... and vice versa... if x decreases from x = b to x = a, the gradient will increase?

i.e. as x increases along the interval, the gradient will fall? ... and vice versa... if x decreases from x = b to x = a, the gradient will increase?

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(Original post by

and are actually reflections in the y-axis, but otherwise this is correct.

You're really just asking questions about whether is increasing or not.

Clearly, if then the gradient is increasing (in the direction of x increasing, hence decreasing in the direction of x decreasing).

And if then the gradient is decreasing (in the dir. of x increasing, hence increasing in the direction of x decreasing).

Therefore, in the domain where we have , then the gradient is indeed the gradient is increasing as x decreases, and vice versa.

Yes. Try to think of this in terms of above as it simplifies the notation a bit and then you don't confuse second derivatives with the notion of increasing/decreasing for the actual function.

**RDKGames**)and are actually reflections in the y-axis, but otherwise this is correct.

You're really just asking questions about whether is increasing or not.

Clearly, if then the gradient is increasing (in the direction of x increasing, hence decreasing in the direction of x decreasing).

And if then the gradient is decreasing (in the dir. of x increasing, hence increasing in the direction of x decreasing).

Therefore, in the domain where we have , then the gradient is indeed the gradient is increasing as x decreases, and vice versa.

Yes. Try to think of this in terms of above as it simplifies the notation a bit and then you don't confuse second derivatives with the notion of increasing/decreasing for the actual function.

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