What is the range of this function?

Question

Let f(x) = 2x^2-2 for 2 < x < 5
_ _

Those are less than and equal to and greater than and equal to signs. So how do you find the range in interval notation. I tried graphing it but it just confused me. Thanks.

Answer 1

Your function 2x^2-2 is an increasing function over the interval [2,5]. If you are familiar with calculus, you can see this by taking the derivative of your function and observing that derivative gives you a positive value if you plug in a number from [2,5]. Knowing the inc/dec property helps as now you know that the smallest value your function can give is the value it obtains at the leftmost point "2" which produces 6 and the highest value is at "5" which is 48. As your function is a continuous one, the answer is all the values between 6 and 48, or [6,48].

Answer 2

let us find f'(x) = 2x
in the range 2 so this is monotomically increasing
at x =2 f(x) = 6
at x = 5 f(x) =21
so the range is 6 < r < 21 or (6,21) (note the ( means do not include)
if 2 <=x < 5
then range is [6 21). 6 inlcuded and 21 excluded
if 2 < x <=5 the (6 ,21]
if 2<=x<=5 the [6,21] both are included

Answer 3

f(2) = 6
f(5) = 48

f(x) is increasing over the entire interval so...
6
The range is an open interval, not a closed one (6,48) not [6,48].
6 and 48 should not be included!

Answer 4

f(2) < f(x)< f(5)
f(2) = 6
f(5)= 48
[ 6, 48]

Answer 5

It is free range, like laying hens.