Does anyone know a good way of finding the range of a function...I usually either man it out and look for the wholes and check its horizontal endpoints...or if its a simple function I find its inverse and check that functions domain
Yeah that is how I check the possibility for a function to not have an element of the range...but it dose not help me for finding the range...for example I could use that method to check whether or not 1 is in the range by saying $\displaystyle 1=f(x)$ and seeing if there is an x value that satisfies that equation...but it doesnt help me to go o well since $\displaystyle f(x)=randomfunction$ then the range is this
Here is an example. Let $\displaystyle y=2e^{-x^2}$, the domain is of course all numbers. The range are all $\displaystyle y$ so that we can solve the equation $\displaystyle y=2e^{-x^2}$ for $\displaystyle x$ in the domain. We begin by writing, $\displaystyle (y/2) = e^{-x^2}$ but that can only have a solution if $\displaystyle (y/2) > 0$ (because the exponentinal is positive). Which means if $\displaystyle y/2>0$ then $\displaystyle \ln (y/2) = -x^2 \implies - \ln (y/2) = x^2$. Now to solve this equation we require $\displaystyle - \ln (y/2) \geq 0 \implies \ln (y/2) < 0 \implies 0<y/2\leq 1\implies 0 < y\leq 2$. For those values we can solve the equation.
Try this example. Find the range of $\displaystyle y = x/(x^2+1)$.