SOLUTION: Find the domain and range of the function f(x) =x ^2 - 2x-3/x^2+2x+3 and hence find expression for f(3x+1) and determine f(4)

Algebra ->  Functions -> SOLUTION: Find the domain and range of the function f(x) =x ^2 - 2x-3/x^2+2x+3 and hence find expression for f(3x+1) and determine f(4)      Log On


   

Question 978054: Find the domain and range of the function f(x) =x ^2 - 2x-3/x^2+2x+3 and hence find expression for f(3x+1) and determine f(4)
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
, and the denominator has no real roots and it positive everywhere. Zeros of f(x) are at x=-1 and x=3.

-infinity to -1,
f is positive.

-1 to +3,
f is negative.

+3 to infinity,
f is positive.


There is a local minimum somewhere between x at -1 and x at 3. A quick guess might be at or near x=2, meaning this minimum might be -2%2F%284%2B4%2B3%29=-2%2F11. That means, as a guess, range is -2/11 to ... unsure the maximum without trying some taking of derivatives. (Not very good a guess).

Here is the graph:
graph%28300%2C300%2C-6%2C6%2C-8%2C8%2C%28x%5E2-2x-3%29%2F%28x%5E2%2B2x%2B3%29%29

Some bad guessing having been done as described above, calculus derivative would have been a better method choice. Actual range based on viewing the graph is -1 to +2 for f(x).