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...