# Thread: Finding range and function of a specific interval

1. ## Finding range and function of a specific interval

Is there an algebraic way to find this if the problem states: Define a function h by $h(x)=f(2x)$ where f is the function defined by $f(x)=x^2$, with the domain of f in the interval [-1,1].

a. Find the domain of h
b. Find the range of h

How would I find the range of h algebraically without creating a table?

2. ## Re: Finding range and function of a specific interval

Is h(4) defined? No - that would have to be = f(2(4)) = f(8), but f is only defined on [-1,1].
Is h(2) defined? No - that would have to be = f(2(2)) = f(4), but f is only defined on [-1,1].
Is h(1) defined? No - that would have to be = f(2(1)) = f(2), but f is only defined on [-1,1].
Is h(1/2) defined? Yes - that would have to be = f(2(1/2)) = f(1) (= 1), since f IS defined on [-1,1].
Is h(1/4) defined? Yes - that would have to be = f(2(1/4)) = f(1/2) (= 1/4), since f IS defined on [-1,1].
Any guesses now what the domain of h might be?
Once you know the domain of h, how to find the range of h will hopefully become a little clearer.

3. ## Re: Finding range and function of a specific interval

the domain of h is the same as the domain of f, but how do I find the range? Can't I just plug in the intervals?

4. ## Re: Finding range and function of a specific interval

Also, I forgot to state; I have to find the interval of the domain and range.

5. ## Re: Finding range and function of a specific interval

I presume that you are able to calculate directly that $h(x)= f(2x)= (2x)^2= 4x^2$. And it should not be difficult to see that h has no negative values while h(0)= 0. Further, f(1)= f(-1)= 4 while for any x with |x|< 1, $x^2< 1$ and so $4x^2< 4$

6. ## Re: Finding range and function of a specific interval

My textbook states that the domain interval for h is: $[-1/2,1/2]$ and the range is in the interval $[0,1]$. Why is this?

7. ## Re: Finding range and function of a specific interval

I mean, the only way that can happen is if $-1 = 2x$ and $1 = 2x$

8. ## Re: Finding range and function of a specific interval

Originally Posted by Eraser147
My textbook states that the domain interval for h is: $[-1/2,1/2]$ and the range is in the interval $[0,1]$. Why is this?
Look at my previous post(#2).
If you look at the examples, and think about them, I suspect that the reason h's domain is what it is might become clear.