DP Mathematics HL Questionbank

• Syllabus

2.1

Sub sections and their related questions

Concept of function \(f:x \mapsto f\left( x \right)\) : domain, range; image (value)

• 12N.1.hl.TZ0.12d: (i)     State \({F_n}(0){\text{ and }}{F_n}(1)\) . (ii)     Show that \({F_n}(x) < x\) ,...
• 09N.1.hl.TZ0.4: Consider the function f , where \(f(x) = \arcsin (\ln x)\). (a)     Find the domain of f . (b)...
• 13M.1.hl.TZ1.12d: Find the range of f.
• 11N.1.hl.TZ0.9b: Hence determine the range of the function \(f:x \to \frac{{x + 1}}{{{x^2} + x + 1}}\).
• 11M.1.hl.TZ1.8b: Find the coordinates of the point where the graph of \(y = f(x)\) and the graph of...
• 11M.1.hl.TZ1.8a: (i)     Find \(\left( {g \circ f} \right)\left( x \right)\) and write down the domain of the...
• 11M.1.hl.TZ1.10a: Find the largest possible domain of the function \(g\) .
• 09N.2.hl.TZ0.9: (a)     Given that the domain of \(g\) is \(x \geqslant a\) , find the least value of \(a\) such...
• 15M.1.hl.TZ1.6b: Given that \(f(x)\) can be written in the form \(f(x) = A + \frac{B}{{2x - 1}}\), find the values...
• 15M.1.hl.TZ1.9b: Find the range of \(g \circ f\).
• 15M.1.hl.TZ2.13a: Show that \(\frac{1}{{\sqrt n  + \sqrt {n + 1} }} = \sqrt {n + 1}  - \sqrt n \) where...
• 15M.1.hl.TZ2.13b: Hence show that \(\sqrt 2  - 1 < \frac{1}{{\sqrt 2 }}\).
• 15N.1.hl.TZ0.12d: Find the range of \(f\).
• 15N.2.hl.TZ0.12a: The functions \(u\) and \(v\) are defined as \(u(x) = x - 3,{\text{ }}v(x) = 2x\) where...
• 16M.2.hl.TZ1.2a: Express \({x^2} + 4x - 2\) in the form \({(x + a)^2} + b\) where \(a,{\text{ }}b \in \mathbb{Z}\).
• 16M.2.hl.TZ1.2b: If \(f(x) = x + 2\) and \((g \circ f)(x) = {x^2} + 4x - 2\) write down \(g(x)\).
• 16M.2.hl.TZ1.5a: Prove that \(f\) is an even function.
• 16M.2.hl.TZ1.5b.i: Sketch the graph \(y = f(x)\).
• 16M.2.hl.TZ1.5b.ii: Write down the range of \(f\).
• 16N.2.hl.TZ0.2: Find the acute angle between the planes with equations \(x + y + z = 3\) and \(2x - z = 2\).
• 17N.1.hl.TZ0.11a: Determine whether \({f_n}\) is an odd or even function, justifying your answer.
• 18M.1.hl.TZ2.10a: Find the inverse function \({f^{ - 1}}\), stating its domain.
• 18M.1.hl.TZ2.10c: The function \(h\) is defined by \(h\left( x \right) = \sqrt x \), for \(x\) ≥ 0. State the...
• 18M.2.hl.TZ2.10a.i: Sketch the graph of \(y = f\left( x \right)\)...
• 18M.2.hl.TZ2.10a.ii: With reference to your graph, explain why \(f\) is a function on the given domain.
• 18M.2.hl.TZ2.10a.iii: Explain why \(f\) has no inverse on the given domain.
• 18M.2.hl.TZ2.10a.iv: Explain why \(f\) is not a function for...

Odd and even functions.

• SPNone.1.hl.TZ0.5a: Determine whether f is even, odd or neither even nor odd.
• 14M.1.hl.TZ2.14d: Nigel states that \(f\) is an odd function and Tom argues that \(f\) is an even function. (i)  ...
• 15M.1.hl.TZ1.5a: Given that \(f\) is an even function, show that \(b = 0\).
• 15M.1.hl.TZ1.5b: Given that \(g\) is an odd function, find the value of \(r\).
• 15M.1.hl.TZ1.5c: The function \(h\) is both odd and even, with domain \(\mathbb{R}\). Find \(h(x)\).
• 15N.1.hl.TZ0.12a: Show that \(f\) is an odd function.
• 18M.1.hl.TZ2.10a: Find the inverse function \({f^{ - 1}}\), stating its domain.
• 18M.1.hl.TZ2.10c: The function \(h\) is defined by \(h\left( x \right) = \sqrt x \), for \(x\) ≥ 0. State the...

Composite functions \(f \circ g\) .

• 12M.1.hl.TZ2.11b: Find an expression for the composite function \(f \circ g(x)\) in the form...
• 12N.1.hl.TZ0.12a: Find an expression for \((f \circ f)(x)\) .  
• 08M.1.hl.TZ1.8: The functions f and g are defined as: \[f(x) = {{\text{e}}^{{x^2}}},{\text{ }}x \geqslant...
• 08M.1.hl.TZ2.4: Let \(f(x) = \frac{4}{{x + 2}},{\text{ }}x \ne - 2{\text{ and }}g(x) = x - 1\). If...
• 13M.2.hl.TZ1.13d: (i)     With f and g as defined in parts (a) and (b), solve \(g \circ f(x) = 2\). (ii)     Let...
• 10M.1.hl.TZ1.2: Shown below are the graphs of \(y = f(x)\) and \(y = g(x)\).     If \((f \circ g)(x) = 3\),...
• 14M.2.hl.TZ1.12: Let \(f(x) = \left| x \right| - 1\). (a)     The graph of \(y = g(x)\) is drawn below.      ...
• 14M.1.hl.TZ2.14b: Find an expression for the composite function \(h \circ g(x)\) and state its domain.
• 15M.1.hl.TZ1.9a: Show that \(g \circ f(x) = 3\sin \left( {2x + \frac{\pi }{5}} \right) + 4\).
• 15M.1.hl.TZ1.9b: Find the range of \(g \circ f\).
• 15M.1.hl.TZ1.9c: Given that \(g \circ f\left( {\frac{{3\pi }}{{20}}} \right) = 7\), find the next value of \(x\),...
• 15M.1.hl.TZ2.11a: Find an expression for \(g \circ f(x)\), stating its domain.
• 15M.1.hl.TZ2.11b: Hence show that \(g \circ f(x) = \frac{{\sin x + \cos x}}{{\sin x - \cos x}}\).
• 15N.2.hl.TZ0.12a: The functions \(u\) and \(v\) are defined as \(u(x) = x - 3,{\text{ }}v(x) = 2x\) where...
• 16N.2.hl.TZ0.2: Find the acute angle between the planes with equations \(x + y + z = 3\) and \(2x - z = 2\).
• 17N.1.hl.TZ0.11a: Determine whether \({f_n}\) is an odd or even function, justifying your answer.
• 18M.1.hl.TZ2.10a: Find the inverse function \({f^{ - 1}}\), stating its domain.
• 18M.1.hl.TZ2.10c: The function \(h\) is defined by \(h\left( x \right) = \sqrt x \), for \(x\) ≥ 0. State the...
• 18M.2.hl.TZ2.10a.i: Sketch the graph of \(y = f\left( x \right)\)...
• 18M.2.hl.TZ2.10a.ii: With reference to your graph, explain why \(f\) is a function on the given domain.
• 18M.2.hl.TZ2.10a.iii: Explain why \(f\) has no inverse on the given domain.
• 18M.2.hl.TZ2.10a.iv: Explain why \(f\) is not a function for...

Identity function.

• 18M.1.hl.TZ2.10a: Find the inverse function \({f^{ - 1}}\), stating its domain.
• 18M.1.hl.TZ2.10c: The function \(h\) is defined by \(h\left( x \right) = \sqrt x \), for \(x\) ≥ 0. State the...

One-to-one and many-to-one functions.

• 09M.1.hl.TZ2.11: A function is defined as \(f(x) = k\sqrt x \), with \(k > 0\) and \(x \geqslant 0\) . (a)    ...
• 15M.2.hl.TZ1.6: A function \(f\) is defined by \(f(x) = {x^3} + {{\text{e}}^x} + 1,{\text{ }}x \in \mathbb{R}\)....
• 18M.1.hl.TZ2.10a: Find the inverse function \({f^{ - 1}}\), stating its domain.
• 18M.1.hl.TZ2.10c: The function \(h\) is defined by \(h\left( x \right) = \sqrt x \), for \(x\) ≥ 0. State the...

Inverse function \({f^{ - 1}}\), including domain restriction. Self-inverse functions.

• 12M.1.hl.TZ2.11c: (i)     Find an expression for the inverse function \({f^{ - 1}}(x)\) . (ii)     State the...
• 12N.1.hl.TZ0.12c: Show that \({F_{ - n}}(x)\) is an expression for the inverse of \({F_n}\) .
• 08M.1.hl.TZ1.8: The functions f and g are defined as: \[f(x) = {{\text{e}}^{{x^2}}},{\text{ }}x \geqslant...
• 08M.1.hl.TZ2.4: Let \(f(x) = \frac{4}{{x + 2}},{\text{ }}x \ne - 2{\text{ and }}g(x) = x - 1\). If...
• 11M.1.hl.TZ2.8: A function is defined by...
• 09N.1.hl.TZ0.4: Consider the function f , where \(f(x) = \arcsin (\ln x)\). (a)     Find the domain of f . (b)...
• SPNone.1.hl.TZ0.13c: Obtain expressions for the inverse function \({f^{ - 1}}\) and state their domains.
• 10M.1.hl.TZ2.10: A function f is defined by \(f(x) = \frac{{2x - 3}}{{x - 1}},{\text{ }}x \ne 1\). (a)     Find...
• 10N.1.hl.TZ0.9: Consider the function \(f:x \to \sqrt {\frac{\pi }{4} - \arccos x} \). (a)     Find the largest...
• 13M.1.hl.TZ2.12d: (i)     Find an expression for \({f^{ - 1}}(x)\). (ii)     Sketch the graph of \(y = f(x)\),...
• 11N.1.hl.TZ0.9c: Explain why f has no inverse.
• 11N.2.hl.TZ0.8a: find \({f^{ - 1}}(x)\), stating its domain;
• 09N.2.hl.TZ0.9: (a)     Given that the domain of \(g\) is \(x \geqslant a\) , find the least value of \(a\) such...
• 14M.2.hl.TZ2.7b: Find the inverse function \({f^{ - 1}}\), stating its domain.
• 14N.1.hl.TZ0.11a: (i)     Find \({f^{ - 1}}(x)\). (ii)     State the domain of \({f^{ - 1}}\).
• 15M.1.hl.TZ1.6a: Find an expression for \({f^{ - 1}}(x)\).
• 15M.1.hl.TZ2.10b: Find an expression for \({f^{ - 1}}(x)\).
• 15M.1.hl.TZ2.10c: Find all values of \(x\) for which \(f(x) = {f^{ - 1}}(x)\).
• 15N.2.hl.TZ0.12b: (i)     Explain why \(f\) does not have an inverse. (ii)     The domain of \(f\) is restricted...
• 15N.2.hl.TZ0.12c: Consider the function defined by \(h(x) = \frac{{2x - 5}}{{x + d}}\), \(x \ne  - d\) and...
• 16M.2.hl.TZ1.11c.i: Write down the largest value of \(a\) for which \(f\) has an inverse. Give your answer correct to...
• 16M.2.hl.TZ1.11c.ii: For this value of a sketch the graphs of \(y = f(x)\) and \(y = {f^{ - 1}}(x)\) on the same set...
• 16M.2.hl.TZ1.11c.iii: Solve \({f^{ - 1}}(x) = 1\).
• 16M.2.hl.TZ1.11d.i: Find an expression for \({g^{ - 1}}(x)\), stating the domain.
• 16M.2.hl.TZ1.11d.ii: Solve \(({f^{ - 1}} \circ g)(x) < 1\).
• 16M.2.hl.TZ2.5: The function \(f\) is defined as...
• 16N.2.hl.TZ0.10d: Use your answers from parts (b) and (c) to justify that \(f\) has an inverse and state its domain.
• 16N.2.hl.TZ0.10e: Find an expression for \({f^{ - 1}}(x)\).
• 17N.1.hl.TZ0.11a: Determine whether \({f_n}\) is an odd or even function, justifying your answer.
• 18M.1.hl.TZ2.10a: Find the inverse function \({f^{ - 1}}\), stating its domain.
• 18M.1.hl.TZ2.10c: The function \(h\) is defined by \(h\left( x \right) = \sqrt x \), for \(x\) ≥ 0. State the...
• 18M.2.hl.TZ2.10a.i: Sketch the graph of \(y = f\left( x \right)\)...
• 18M.2.hl.TZ2.10a.ii: With reference to your graph, explain why \(f\) is a function on the given domain.
• 18M.2.hl.TZ2.10a.iii: Explain why \(f\) has no inverse on the given domain.
• 18M.2.hl.TZ2.10a.iv: Explain why \(f\) is not a function for...