Answer to Question #294830 in Calculus for Sherlock

Question

Domain and renge f(x)=1/2x-p

Answer 1

Let us find the domain and range of $f(x)=\frac{1}{2x-p}.$

It follows that the domain is equal to the set $\R\setminus\{\frac{p}2\}.$

If $y=0$ then the equation $y=\frac{1}{2x-p}$ has no solution. If $y\ne 0$ then the equation $y=\frac{1}{2x-p}$ is equivalent to $2x-p=\frac{1}{y}$, and hence has a solution $x=\frac{1}{2y}+\frac{p}2$ . Therefore, $f(x)=f(\frac{1}{2y}+\frac{p}2)=y.$ We conclude that the range of the function $f$ is equal to the set $\R\setminus\{0\}.$