Hi there how would you solve this question algebratically:

Find the range of this function:

f(x)=1-3sin(2x), 0<x<Pi/2

First i multiply the restriction by 2

0<2x<Pi

Then I don't know how to input the sine part in
Thanks

jiten55

When A is between 0 an pi, Sin A is between 0 and 1

Hence least value of Sin A = 0, Greatest = 1

Hence least value of -SinA = -1, Greatest = 0

Hence least value of -3 SinA = -3, Greatest = 0

Hence least value of 1-3sin(2x) = 1 - 3 = -2, Greatest = 1

Hence Range is [-2, 1] (Closed interval)

PolluxCastor

You can't find the range in ALL cases by substituting into the function the end points of the domain. You need to find what the minimum and maximum the function will output given the domain.

The function has a minimum of .75 at x = -.5

If we limit the domain to [0, 2] the range could be found by substituting the end points.
However, if the stated domain were [-1, 1], substituting the end points would give the result [1, 3], but the actual range is [.75, 3]

You can use the domain as a starting point, then determine if there is a minima or maxima that would be included.

I hope that helped.

Edited function to what I graphed, Thanks Jiten.

jiten55

I did not substitute the end points of the domain! At the end points of the domain, values of the function are simply 1 at both points! If I substituted the end points of the domain, the result would have been [1, 1] which is ridiculous.

I used the domain to determine greatest and least values of the function in the domain.

That is absolutely essential.

I do not think finding maxima or minima will help in this case.

What we need in this case is Least and Greatest.

My answer addressed only this particular question, so that the answer is easy to understand.

However, it applies, I believe, in case of all functions. As long as you determine the least and greatest value of the function in the specified domain.

NB The function x^2 + 2x + 1 does not have a minimum at x = -.5

It has a minimum only at x = -1

The function x^2 + 2x + 1 has range [0, 4] in the domain [-1, 1] and this range is correctly obtained by substituting x = -1 and x = 1.

x^2 + 2x + 1 = (x + 1)^2 and hence greatest and least values are easily determined.

In case of f(x) = x^2 + x + 1 = (x + .5)^2 + .75

and we can easily find least and greatest values in any domain.

Find least and greatest of (x + .5)^2 and add .75 to each!

jiten55

Furthermore, we were to "solve this question algebratically".

Not only Maxima/Minima are not needed in this case, their determination requires Calculus!

CONCLUSION:

To determine Range of f(x) : Determine the least value and the greatest value of f(x) when x lies in the domain.

PERIOD.

Range of the function, by definition, is between the above 2 values.

In more complicated functions, when Calculus is permitted, use the following procedure to obtain the greatest and the least values of f(x).

1. Find minimum and maximum values of function in the domain.

2. Find values of the function at end points of the domain.

3. Greatest value = Greatest of the above values. (in step 1 and 2)

4. Least value = Least of the above values (in step 1 and 2)

5. Range = [Least value, Greatest Value]

PolluxCastor

I was answering the original question, I had not seen your post (Jiten, btw, hi) until after I posted.

I was pointing out the extremes of the domain and range may not occur at the same point in the function.

I did not use calculus to determine the maxima and minima, I did that graphicaly with excel.

Sorry to step on your toes.