Undergraduate algebra problem of field theory

Question

I tried to solve but I couldn't. Can you solve this problem?

1. Let $F$ be a field and let $t$ be transcendental over $F$. Let $x\in F(t)$ and suppose $x\not\in F$. Prove that $F(t)$ is algebraic over $F(x)$.
2. If $x$ is expressed as a quotient of relatively prime polynomials, i.e. $x = \dfrac{f(t)}{g(t)}$, find $[F(t):F(x)]$ in terms of degrees of $f$ and $g$. Prove all assertions.

Using hint , i could solve it like below.Is it correct?

About problem 1

Let $h(X)=\dfrac{f(t)}{g(t)}g(X)-f(X)\in F(x)[X].$

then $h(t)=0.$

then $t$ is algebraic number over $F(x)$.

then $[F(t):F(x)]$ is finite dimension.

then $F(t)$ is algebraic over $F(x)$.

About problem 2

$h(X)$is minimal polinomial of $t$ over $F(x)$.

then $deg(h(X))=[F(t):F(x)]$

onthe other hand $deg(h(X))=max\{deg(f(X)),deg(g(X))\}$

then $[F(t):F(x)]=max\{deg(f(X)),deg(g(X))\}$