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...