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2021 Paper 2 Q4
D: 1500.0 B: 1500.0
curve sketching inverse functions exponentials and logarithms lambert w function stationary points domain and range

  1. Sketch the curve \(y = xe^x\), giving the coordinates of any stationary points.
  2. The function \(f\) is defined by \(f(x) = xe^x\) for \(x \geqslant a\), where \(a\) is the minimum possible value such that \(f\) has an inverse function. What is the value of~\(a\)? Let \(g\) be the inverse of \(f\). Sketch the curve \(y = g(x)\).
  3. For each of the following equations, find a real root in terms of a value of the function~\(g\), or demonstrate that the equation has no real root. If the equation has two real roots, determine whether the root you have found is greater than or less than the other root.
    1. \(e^{-x} = 5x\)
    2. \(2x \ln x + 1 = 0\)
    3. \(3x \ln x + 1 = 0\)
    4. \(x = 3\ln x\)
  4. Given that the equation \(x^x = 10\) has a unique positive root, find this root in terms of a value of the function~\(g\).

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2013 Paper 1 Q8
D: 1516.0 B: 1474.0
functions composite functions domain and range curve sketching asymptotes logarithm square root inverse functions

  1. The functions \(\mathrm{a, b, c}\) and \(\mathrm{d}\) are defined by
    • \({\rm a}(x) =x^2 \ \ \ \ (-\infty < x < \infty),\)
    • \({\rm b}(x) = \ln x \ \ \ \ (x > 0),\)
    • \({\rm c}(x) =2x \ \ \ \ (-\infty < x < \infty),\)
    • \({\rm d}(x)= \sqrt x \ \ \ \ (x\ge0) \,.\)
    Write down the following composite functions, giving the domain and range of each: \[ \rm cb, \quad ab, \quad da, \quad ad. \]
  2. The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by
    • \(\f(x)= \sqrt{x^2-1\,} \ \ \ \ (\vert x \vert \ge 1),\)
    • \(\g(x) = \sqrt{x^2+1\,} \ \ \ \ (-\infty < x < \infty).\)
    Determine the composite functions \(\mathrm{fg}\) and \(\mathrm{gf}\), giving the domain and range of each.
  3. Sketch the graphs of the functions \(\h\) and \({\rm k}\) defined by
    • \(\h(x) = x+\sqrt{x^2-1\,}\, \ \ \ \ ( x \ge1)\),
    • \({\rm k}(x) = x-\sqrt{x^2-1\,}\, \ \ \ \ (\vert x\vert \ge1),\)
    justifying the main features of the graphs, and giving the equations of any asymptotes. Determine the domain and range of the composite function \(\mathrm{kh}\).

Solution:

  1. \begin{align*} cb(x) &= c(b(x)) \\ &= 2 \ln x \quad (x > 0) \\ ab(x) &= (b(x))^2 \\ &= (\ln x)^2 \quad (x > 0) \\ da(x) &= \sqrt{a(x)} \\ &= \sqrt{x^2} \\ &= |x| \quad (-\infty < x < \infty) \\ ad(x) &= (d(x))^2 \\ &= (\sqrt{x})^2 \\ &= x \quad (x \geq 0) \end{align*} The domains are specified above. The ranges are \(\mathbb{R}, \mathbb{R}_{\geq 0}, \mathbb{R}_{\geq 0}, \mathbb{R}_{\geq 0}\) respectively.
  2. \begin{align*} fg(x) &= \sqrt{g(x)^2-1} \quad (|g(x)| \geq 1) \\ &= \sqrt{x^2+1-1} \\ &= |x| \end{align*} So \(fg: \mathbb{R} \to \mathbb{R}_{\geq 0}\). \begin{align*} gf(x) &= \sqrt{f(x)^2 + 1} \\ &= \sqrt{\left ( \sqrt{x^2-1} \right)^2+1} \quad (|x| \geq 1) \\ &= |x| \end{align*} So \(gf: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}\)
    • TikZ diagram
    • TikZ diagram
    \begin{align*} kh(x) &= h(x) - \sqrt{h(x)^2 -1} \quad (|h(x)| \geq 1)\\ &= x + \sqrt{x^2+1} - \sqrt{(x + \sqrt{x^2+1})^2 - 1} \\ &= x + \sqrt{x^2+1} - \sqrt{x^2 + x^2 - 2x} \quad (x \geq 1) \\ &= x + \sqrt{x^2+1} - \sqrt{2x^2-2x} \quad (x \geq 1) \end{align*} This has domain \(x \geq 1\) and range, \((0, 1]\)

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2001 Paper 3 Q4
D: 1700.0 B: 1473.9
trigonometry inverse trigonometric functions curve sketching substitution arcsin arctan domain and range rational function

In this question, the function \(\sin^{-1}\) is defined to have domain \( -1\le x \le 1\) and range \linebreak \( - \frac{1}{2}\pi \le x \le \frac{1}{2}\pi\) and the function \(\tan^{-1}\) is defined to have the real numbers as its domain and range \( - \frac{1}{2}\pi < x < \frac{1}{2}\pi\).

  1. Let $$ \g(x) = \displaystyle {2x \over 1 + x^2}\;, \ \ \ \ \ \ \ \ \ \ -\infty
  2. Let \[ \displaystyle \f \l x \r = \sin^{-1} \l {2x \over 1 + x^2} \r \;,\ \ \ \ \ \ \ \ \ -\infty < x < \infty\;. \] Show that $ \f(x ) = 2 \tan^{-1} x\( for \) -1 \le x \le 1\,\( and \)\f(x) = \pi - 2 \tan^{-1} x \( for \)x\ge1\,$. Sketch the graph of \(\f(x)\).

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1996 Paper 3 Q8
D: 1700.0 B: 1516.0
simultaneous equations Möbius transformation inverse function composition of functions fixed points quadratic equation rational function domain and range

A transformation \(T\) of the real numbers is defined by \[ y=T(x)=\frac{ax-b}{cx-d}\,, \] where \(a,b,c\), \(d\) are real numbers such that \(ad\neq bc\). Find all numbers \(x\) such that \(T(x)=x.\) Show that the inverse operation, \(x=T^{-1}(y)\) expressing \(x\) in terms of \(y\) is of the same form as \(T\) and find corresponding numbers \(a',b',c'\),\(d'\). Let \(S_{r}\) denote the set of all real numbers excluding \(r\). Show that, if \(c\neq0,\) there is a value of \(r\) such that \(T\) is defined for all \(x\in S_{r}\) and find the image \(T(S_{r}).\) What is the corresponding result if \(c=0\)? If \(T_{1},\) given by numbers \(a_{1},b_{1},c_{1},d_{1},\) and \(T_{2},\) given by numbers \(a_{2},b_{2},c_{2},d_{2}\) are two such transformations, show that their composition \(T_{3},\) defined by \(T_{3}(x)=T_{2}(T_{1}(x)),\) is of the same form. Find necessary and sufficient conditions on the numbers \(a,b,c,d\) for \(T^{2}\), the composition of \(T\) with itself, to be the identity. Hence, or otherwise, find transformations \(T_{1},T_{2}\) and their composition \(T_{3}\) such that \(T_{1}^{2}\) and \(T_{2}^{2}\) are each the identity but \(T_{3}^{2}\) is not.

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