If x is real, show that each of the following expression is capabale of assuming all real values: (i) (2x^2+4x+1)/(x^2+4x+2) (ii) p^2/(1-x)-q^2/(1+x)?
#(i) (2x^2+4x+1)/(x^2+4x+2) (ii) p^2/(1-x)-q^2/(1+x)#
1 Answer
I'm going to do this a little theoretically; I hope that's okay.
We can only see the sign of the function switch if we have an asymptote or a zero, so let's find those.
For the first, we find the vertical asymptotes, i.e. where the denominator is 0:
The actual values aren't that important past the first decimal.
The zeroes are when the numerator is 0:
So we need to consider the signs in the following regions:
If either end of the region is an asymptote, it will go to
Therefore, we can just check some numbers in these ranges:
(So the function goes to
(So the function goes to
All of the above work was being really careful and most likely you don't have to be so careful.
For the second equation, we can simplify it into one statement
So, we know that the slope on the numerator is positive since it is the sum of two squares and the denominator has asymptotes at
As the function approaches
As the function approaches
Therefore, both functions can take on any real value.