Question

Let $f:X\to R,g:Y\to R$ where $X$ and $Y$ are non-empty subsets of $R$. Suppose also that for all $xϵX$, the compositions are defined and $\left(fog\right)\left(x\right)=\left(gof\right)\left(x\right)$. How many of the following five statements must be true?
(i) $X\subset Y$
(ii) $Y\subset X$
(iii) $f$ and $g$ are inverse
(iv) both $f$ and $g$ are one - to - one
(v) both $f$ and $g$ are continuous.

• A. $4$
• B. $3$
• C. $2$
• D. $1$

Answer 1

The correct option is D $1$

(i) $X\subset Y$ True : This must be the case since the composition $\left(fog\right)\left(x\right)$ is defined all $xϵX$ just also be in $Y$ which is the definition of subset.
(ii) $Y\subset X$ False: Take $f\left(x\right)=g\left(x\right)=x$, where $X=[0,1]$ and $Y[0,2]$ as a counter example.
(iii) $f$ and $g$ are inverse false: Take the same example as above. To be inverse, the composition must work for all $x$ in the domains of each function, but the composition $\left(fog\right)\left(x\right)$ is not defined for $xϵ(1,2]$.
(iv) both $f$ and $g$ are one - to - one false: Take $f\left(x\right)=x,g\left(x\right)=\left[x\right]$ where $\left[x\right]$ is the greatest integer function. Taking $X=Y=R^{+}\cup \left\{0\right\}$, the composition always hold but $g$ is not $1-1$ since it will not pass the horizontal line test.
(v) Both $f$ and $g$ are continuous false: Using the same example as above, we see $g$ is not continuous, yet the compositions hold.