Math CentralQuandaries & Queries


Question from Nazrul, a teacher:

f : R to R is a function which is defined by f(x)=(2x+2)/(x-1).
Is f(x) an onto function? Please explain.
What is the range of this function?
Thank you very much for your previous help.

Hi Nazrul,

Whenever I am asked to find the range of a function I want to see a graph. If you have a graphing calculator or a graphing program on your computer sketch a graph of $f(x)=(2x+2)/(x-1)$. If you do so you will see that there is one value on the Y-axis that doesn't seem to be obtained for any value on the X-axis. Suppose that value is $y = a.$ Using this value, attempt to solve the equation $(2x+2)/(x-1) = a$ for $x.$ You will see that no such value of $x$ is possible.

Now repeat the process above but with an unknown value $y = z.$ That is solve the equation $(2x+2)/(x-1) = z$ for $z.$ You should see that the specific value $z = a$ yields no solution. Does any other value of $z$ yield no solution?


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