Problems

Filters
Clear Filters

2 problems found

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~\(g\).

View
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, \((0, 1]\)

View