Finding the Domain of f with Restrictions for Range (-2,4)U(4,11)

[philippe311]

Homework Statement

For f(x)=((x^(2)-2x-3)/(x-3)), restrict the domain of f(find Dom (f) first) so that Rnage(f)=(-2,4)U(4,11)

The Attempt at a Solution

I found that when x=3, the function will be undefined. so the domain(f) should be (-infinity,3)U(3,+infinity). I GUESS now I should go to the equation again and factor the nomunator so that I get (x-3)(x+1)/(x-3). then f(x)= x+1. as the Range given above. I have to plug it in and solve for x, RIGHT?
-2<x+1<11
-3<x<10 = the domain of f with te restriction that x is Not equal to 3. Is that right OR am I missing something?

 

[mighty2000]

Hello, you are on the right track! However, it is important to note that when simplifying the function to f(x)=x+1, we are also changing the original function. So the domain of the original function will also change.

To find the domain of the original function, we need to consider both the restrictions from the original function and the simplified function.

For the original function, as you mentioned, x cannot equal 3. But for the simplified function, x can also not equal -1, since this would make the denominator 0.

So the domain of the original function would be (-infinity, -1)U(-1, 3)U(3, +infinity). This ensures that both the original and simplified function are defined.

Now, to find the range, we can plug in the domain values into the simplified function. We get f(x) = x+1, so the range would be (-2, 4)U(4, 11).

Hope this helps! Let me know if you have any further questions.