Solve the following two independent problems. (i) Let $f$ be a function from domain $S$ to codomain $T$. Let $g$ be another function from domain $T$ to codomain $U$. For each of the blanks below choose a single letter corresponding to one of the four options listed underneath. (It is not necessary that each choice is used exactly once.) Write your answers as a sequence of four letters in correct order. Do NOT explain your answers. If $g \circ f$ is one-to-one then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ . If $g \circ f$ is onto then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ . Option A: must be one-to-one and must be onto. Option B: must be one-to-one but need not be onto. Option C: need not be one-to-one but must be onto. Option D: need not be one-to-one and need not be onto. Recall: $g \circ f$ is the function defined by $g \circ f ( a ) = g ( f ( a ) )$. The function $f$ is said to be one-to-one if, for any $a _ { 1 }$ and any $a _ { 2 }$ in $S , f \left( a _ { 1 } \right) = f \left( a _ { 2 } \right)$ implies $a _ { 1 } = a _ { 2 }$. The function $f$ is said to be onto if, for any $b$ in $T$, there is an $a$ in $S$ such that $f ( a ) = b$. (ii) In the given figure $ABCD$ is a square. Points $X$ and $Y$, respectively on sides $BC$ and $CD$, are such that $X$ lies on the circle with diameter $AY$. What is the area of the square $ABCD$ if $AX = 4$ and $AY = 5$? (Figure is schematic and not to scale.)
Solve the following two independent problems.
(i) Let $f$ be a function from domain $S$ to codomain $T$. Let $g$ be another function from domain $T$ to codomain $U$. For each of the blanks below choose a single letter corresponding to one of the four options listed underneath. (It is not necessary that each choice is used exactly once.) Write your answers as a sequence of four letters in correct order. Do NOT explain your answers.
If $g \circ f$ is one-to-one then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ .\\
If $g \circ f$ is onto then $f$ $\_\_\_\_$ and $g$ $\_\_\_\_$ .
Option A: must be one-to-one and must be onto.\\
Option B: must be one-to-one but need not be onto.\\
Option C: need not be one-to-one but must be onto.\\
Option D: need not be one-to-one and need not be onto.\\
Recall: $g \circ f$ is the function defined by $g \circ f ( a ) = g ( f ( a ) )$. The function $f$ is said to be one-to-one if, for any $a _ { 1 }$ and any $a _ { 2 }$ in $S , f \left( a _ { 1 } \right) = f \left( a _ { 2 } \right)$ implies $a _ { 1 } = a _ { 2 }$. The function $f$ is said to be onto if, for any $b$ in $T$, there is an $a$ in $S$ such that $f ( a ) = b$.
(ii) In the given figure $ABCD$ is a square. Points $X$ and $Y$, respectively on sides $BC$ and $CD$, are such that $X$ lies on the circle with diameter $AY$. What is the area of the square $ABCD$ if $AX = 4$ and $AY = 5$? (Figure is schematic and not to scale.)