Function Analysis Practice Questions

Question 1:
Given the function $f:\mathbb{R}\to \mathbb{R\; defined\; by f(x}$$f(x) = 3x - 5$

• Determine if $f$ is injective.
• Determine if $f$ is surjective.

Question 2:
Given the function $g: \mathbb{Z} \to \mathbb{Z}$ defined by $g(x) = x^3$

• Determine if $g$ is injective.
• Determine if $g$ is surjective.

Question 3:
Given the function $h: \mathbb{R} \to \mathbb{R}$ defined by $h(x) = e^x$

• Determine if $h$ is injective.
• Determine if $h$ is surjective.

Question 4:
Given the function $k: \mathbb{Q} \to \mathbb{Q}$ defined by $k\left(\frac{p}{q}\right) = \frac{p+q}{q}$

• Determine if $k$ is injective.
• Determine if $k$ is surjective.

Question 5:
Given the function $m: \mathbb{Z} \to \mathbb{Z}$ defined by $m(x) = |x|$:

• Determine if $m$ is injective.
• Determine if $m$ is surjective.

Question 6:
Given the function $n: \mathbb{N} \to \mathbb{N}$ defined by $n(x)=x^2$

• Determine if $n$ is injective.
• Determine if $n$ is surjective.

Question 7:
Given the function $p: \mathbb{R} \to \mathbb{R}$ defined by $p(x)=\cos (x)$

• Determine if $p$ is injective.
• Determine if $p$ is surjective.

Question 8:
Given the function $q: \mathbb{Q} \to \mathbb{Z}$ defined by $q\left(\frac{p}{q}\right) = p \cdot q$

• Determine if $q$ is injective.
• Determine if $q$ is surjective.

Question 9:
Given the function $r: \mathbb{R} \to \mathbb{R}$ defined by $r(x) = x^2 + 1$

• Determine if $r$ is injective.
• Determine if $r$ is surjective.

Question 10:
Given the function $s: \mathbb{Z} \to \mathbb{Z}$ defined by $s(x) = x - 1$ 

• Determine if $s$ is injective.
• Determine if $s$ is surjective.