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AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction

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Topic 2—Functions » AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction

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Directly related questions

21M.2.AHL.TZ2.12a:
Show that $f$ $f$ is an even function.
22M.1.AHL.TZ2.6a:
Show that $f$ $f$ is an odd function.
22M.1.AHL.TZ2.11b.i:
Show that $g^{- 1} (x) = 1 + \frac{\sqrt{4 x^{2} + x}}{x}$ $g^{- 1} (x) = 1 + \frac{\sqrt{4 x^{2} + x}}{x}$ .
22M.1.AHL.TZ2.11b.ii:
State the domain of $g^{- 1}$ $g^{- 1}$ .
22M.2.AHL.TZ1.6a:
Show that $f$ $f$ is an odd function.
22M.2.AHL.TZ1.10b.i:
Show that the inverse function of $f$ $f$ is given by $f^{- 1} (x) = \sqrt{x^{2} + 1}$ $f^{- 1} (x) = \sqrt{x^{2} + 1}$ .
SPM.1.AHL.TZ0.9a:
Find the largest value of $a$ $a$ such that $f$ $f$ has an inverse function.
SPM.1.AHL.TZ0.9b:
For this value of $a$ $a$ , find an expression for $f^{- 1} (x)$ ${f^{ - 1}}\left( x \right)$ , stating its domain.
19M.1.AHL.TZ2.H_11a:
Describe the transformation by which $f (x)$ $f\left( x \right)$ is transformed to $g (x)$ $g\left( x \right)$ .
19M.1.AHL.TZ2.H_11b:
State the range of $g$ $g$ .
19M.1.AHL.TZ2.H_11c:
Sketch the graphs of $y = f (x)$ $y = f\left( x \right)$ and $y = g (x)$ $y = g\left( x \right)$ on the same axes, clearly stating the points of intersection with any axes.
19M.1.AHL.TZ2.H_11d:
Find the coordinates of P.
18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f (x)$ $y = f\left( x \right)$ .
18N.2.AHL.TZ0.H_9c.i:
sketch the graph of $y = f (x)$ $y = f\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
18N.2.AHL.TZ0.H_9c.ii:
sketch the graph of $y = f^{- 1} (x)$ $y = {f^{ - 1}}\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality $f (x) > f^{- 1} (x)$ $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .
19M.2.AHL.TZ1.H_4a:
Write down the range of $f$ $f$ .
19M.2.AHL.TZ1.H_4b:
Find $f^{- 1} (x)$ $f'\left( x \right)$ , stating its domain.
17N.1.AHL.TZ0.H_11a:
Determine whether $f_{n}$ ${f_n}$ is an odd or even function, justifying your answer.
18N.1.AHL.TZ0.H_3a:
For $a = - \frac{π}{2}$ $a = - \frac{\pi }{2}$ , sketch the graph of $y = g (x)$ $y = g\left( x \right)$ . Indicate clearly the maximum and minimum values of the function.
17N.1.AHL.TZ0.H_11c:
Hence or otherwise, find an expression for the derivative of $f_{n} (x)$ ${f_n}(x)$ with respect to $x$ $x$ .
17N.1.AHL.TZ0.H_11b:
By using mathematical induction, prove that

$f_{n} (x) = \frac{sin 2^{n + 1} x}{2^{n} sin 2 x}, x \neq \frac{m π}{2}$ ${f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}$ where $m \in \mathbb{Z}$ .
17N.1.AHL.TZ0.H_11d:
Show that, for $n > 1$ , the equation of the tangent to the curve $y = {f_n}(x)$ at $x = \frac{\pi }{4}$ is $4x - 2y - \pi = 0$ .

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