Let $k$ be an algebraically closed field.

(a) Let $X$ and $Y$ be affine varieties defined over $k$. Given a map $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of affine varieties.

(b) Let $f: \mathbb{A}^{1} \rightarrow \mathbb{A}^{3}$ be the map given by

$$f(t)=\left(t, t^{2}, t^{3}\right)$$

Show that $f$ is a morphism. Show that the image of $f$ is a closed subvariety of $\mathbb{A}^{3}$ and determine its ideal.

(c) Let $g: \mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1} \rightarrow \mathbb{P}^{7}$ be the map given by

$g\left(\left(s_{1}, t_{1}\right),\left(s_{2}, t_{2}\right),\left(s_{3}, t_{3}\right)\right)=\left(s_{1} s_{2} s_{3}, s_{1} s_{2} t_{3}, s_{1} t_{2} s_{3}, s_{1} t_{2} t_{3}, t_{1} s_{2} s_{3}, t_{1} s_{2} t_{3}, t_{1} t_{2} s_{3}, t_{1} t_{2} t_{3}\right) .$

Show that the image of $g$ is a closed subvariety of $\mathbb{P}^{7}$.