Regard $K=\mathbb{R}-\lbrace{0\rbrace}$ as a multiplication group. Let $f:K\to K$ be a multiplication homormorphism.

Question 1. Whether that $f$ is surjective implies that $f$ is injective?

Question 2. Whether that $f$ is injective implies that $f$ is surjective?

Question 3. $g: x\to x^b$ is a multiplication homormorphism of $K$ where $b=n/m, (n,m)=1$,$n\in\mathbb{Z}$ and $m$ is an odd integer. How to find any other multiplication homormorphism of $K$ than this form. Any example?

Edit. Emil Jeřábek gave other explicit examples: $g: x\to |x|^r$ or $x\to sgn(x)|x|^r.$

Any other explicit ones?

Perhaps, these questions look like homework, but not easy to me to answer (my major is not in algebraic theory).

Regard $K=\mathbb{R}-\lbrace{0\rbrace}$ as a multiplication group. Let $f:K\to K$ be a multiplication homormorphism.

Question 1. Whether that $f$ is surjective implies that $f$ is injective?

Question 2. Whether that $f$ is injective implies that $f$ is surjective?

Question 3. $g: x\to x^b$ is a multiplication homormorphism of $K$ where $b=n/m, (n,m)=1$,$n\in\mathbb{Z}$ and $m$ is an odd integer. How to find any other multiplication homormorphism of $K$ than this form. Any example?

Edit. Emil Jeřábek gave other explicit examples: $h: x\to |x|^r$ or $x\to sgn(x)|x|^r.$ Of course, $hg$ is also ok.

Any other explicit ones?

Perhaps, these questions look like homework, but not easy to me to answer (my major is not in algebraic theory).

Regard $K=\mathbb{R}-\lbrace{0\rbrace}$ as a multiplication group. Let $f:K\to K$ be a multiplication homormorphism.

Question 1. Whether that $f$ is surjective implies that $f$ is injective?

Question 2. Whether that $f$ is injective implies that $f$ is surjective?

Question 3. $g: x\to x^b$ is a multiplication homormorphism of $K$ where $b=n/m, (n,m)=1$,$n\in\mathbb{Z}$ and $m$ is an odd integer. How to find any other multiplication homormorphism of $K$ than this form. Any example?

Perhaps, these questions look like homework, but not easy to me to answer (my major is not in algebraic theory).

Regard $K=\mathbb{R}-\lbrace{0\rbrace}$ as a multiplication group. Let $f:K\to K$ be a multiplication homormorphism.

Question 1. Whether that $f$ is surjective implies that $f$ is injective?

Question 2. Whether that $f$ is injective implies that $f$ is surjective?

Question 3. $g: x\to x^b$ is a multiplication homormorphism of $K$ where $b=n/m, (n,m)=1$,$n\in\mathbb{Z}$ and $m$ is an odd integer. How to find any other multiplication homormorphism of $K$ than this form. Any example?

Edit. Emil Jeřábek gave other explicit examples: $g: x\to |x|^r$ or $x\to sgn(x)|x|^r.$

Any other explicit ones?

Perhaps, these questions look like homework, but not easy to me to answer (my major is not in algebraic theory).