DP Mathematics HL Questionbank

• Syllabus

8.3

Sub sections and their related questions

Functions: injections; surjections; bijections.

• 12M.3srg.hl.TZ0.4a: The function $g:\mathbb{Z} \to \mathbb{Z}$ is defined by...
• 12M.3srg.hl.TZ0.4b: The set S is finite. If the function $f:S \to S$ is injective, show that f is surjective.
• 12M.3srg.hl.TZ0.4c: Using the set ${\mathbb{Z}^ + }$ as both domain and codomain, give an example of an...
• 08M.3srg.hl.TZ1.2: (a)     Find the range of f . (b)     Prove that f is an injection. (c)     Taking the codomain...
• 08M.3srg.hl.TZ2.2a: Below are the graphs of the two functions $F:P \to Q{\text{ and }}g:A \to B$ . Determine,...
• 08M.3srg.hl.TZ2.2b: Given two functions $h:X \to Y{\text{ and }}k:Y \to Z$ .  Show that (i)     if both h and k...
• 11M.3srg.hl.TZ0.4: The function...
• 09M.3srg.hl.TZ0.4: (a)     Show that $f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}$ defined by...
• 09N.3srg.hl.TZ0.2: The function $f:[0,{\text{ }}\infty [ \to [0,{\text{ }}\infty [$ is defined by...
• 10M.3srg.hl.TZ0.1: The function $f:\mathbb{R} \to \mathbb{R}$ is defined...
• 10N.3srg.hl.TZ0.2a: Let $f:\mathbb{Z} \times \mathbb{R} \to \mathbb{R},{\text{ }}f(m,{\text{ }}x) = {( - 1)^m}x$....
• 10N.3srg.hl.TZ0.2b: P is the set of all polynomials such that...
• 10N.3srg.hl.TZ0.2c: Let $h:\mathbb{Z} \to {\mathbb{Z}^ + }$,...
• 13M.3srg.hl.TZ0.3a: (i)     Sketch the graph of f. (ii)     By referring to your graph, show that f is a bijection.
• 13M.3srg.hl.TZ0.3b: Find ${f^{ - 1}}(x)$.
• 11N.3srg.hl.TZ0.5a: Show that if both f and g are injective, then $g \circ f$ is also injective.
• 11N.3srg.hl.TZ0.5b: Show that if both f and g are surjective, then $g \circ f$ is also surjective.
• 11N.3srg.hl.TZ0.5c: Show, using a single counter example, that both of the converses to the results in part (a) and...
• 14M.3srg.hl.TZ0.3b: Consider the function $f:X \times Y \to \mathbb{R}$ defined by $f(x,{\text{ }}y) = x + y$ and...
• 13N.3srg.hl.TZ0.1: Consider the following functions     ...
• 14N.3srg.hl.TZ0.2a: Prove that $f$ is an injection.
• 14N.3srg.hl.TZ0.2b: Prove that $f$ is not a surjection.
• 15M.3srg.hl.TZ0.4a: Prove that:  (i)     $f$ is an injection, (ii)     $g$ is a surjection.
• 15M.3srg.hl.TZ0.4b: Given that $X = {\mathbb{R}^ + } \cup \{ 0\}$ and $Y = \mathbb{R}$, choose a suitable pair...
• 16M.3srg.hl.TZ0.4a: Prove that $f$ is an injection.
• 16M.3srg.hl.TZ0.4b: (i)     Prove that $f$ is a surjection. (ii)     Hence, or otherwise, write down the inverse...
• 16N.3srg.hl.TZ0.2a: (i)     Sketch the graph of $y = f(x)$ and hence justify whether or not $f$ is a...
• 16N.3srg.hl.TZ0.2b: (i)     Sketch the graph of $y = g(x)$ and hence justify whether or not $g$ is a...
• 18M.3srg.hl.TZ0.5b.i: Show that $f$ is injective.
• 18M.3srg.hl.TZ0.5b.ii: Show that $f$ is surjective.

Composition of functions and inverse functions.

• 08M.3srg.hl.TZ1.2: (a)     Find the range of f . (b)     Prove that f is an injection. (c)     Taking the codomain...
• 08N.3srg.hl.TZ0.3: Two functions, F and G , are defined on $A = \mathbb{R}\backslash \{ 0,{\text{ }}1\}$...
• 08N.3srg.hl.TZ0.5: Three functions mapping $\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ are defined...
• 09M.3srg.hl.TZ0.4: (a)     Show that $f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}$ defined by...
• 09N.3srg.hl.TZ0.2: The function $f:[0,{\text{ }}\infty [ \to [0,{\text{ }}\infty [$ is defined by...
• 10M.3srg.hl.TZ0.1: The function $f:\mathbb{R} \to \mathbb{R}$ is defined...
• 15M.3srg.hl.TZ0.4a: Prove that:  (i)     $f$ is an injection, (ii)     $g$ is a surjection.
• 15M.3srg.hl.TZ0.4b: Given that $X = {\mathbb{R}^ + } \cup \{ 0\}$ and $Y = \mathbb{R}$, choose a suitable pair...