Math CentralQuandaries & Queries


Question from Ariana, a student:

If f o g are one to one function,does it follow that g is one to one? Give reasons for your answers

Hi Ariana,

I think your problem should say

If f o g is a one to one function, does it follow that g is one to one?

A function $h$ is one to one if it is true that for any $x_1$ and $x_2$ in the domain of $h,$ if $h(x_1) = h(x_2)$ then $x_1 = x_2.$

Suppose that $x_1$ and $x_2$ are in the domain of $g$ and that $g(x_1) = g(x_2).$ Since $g(x_1) = g(x_2)$ and $f$ is a function, $f(g(x_1)) = f(g(x_2)),$ that is $f \circ g(x_1) = f \circ g(x_2).$ What do you know about $x_1$ and $x_2?$


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