1.02m Graphs of functions: difference between plotting and sketching

The function \(f\) is such that \(f(x) = 3 - 4\cos^k x\), for \(0 \leq x \leq \pi\), where \(k\) is a constant.

1. In the case where \(k = 2\),
   1. find the range of \(f\), [2]
   2. find the exact solutions of the equation \(f(x) = 1\). [3]
2. In the case where \(k = 1\),
   1. sketch the graph of \(y = f(x)\), [2]
   2. state, with a reason, whether \(f\) has an inverse. [1]

Functions \(f\) and \(g\) are defined by $$f : x \mapsto 2x + 5 \quad \text{for } x \in \mathbb{R},$$ $$g : x \mapsto \frac{8}{x - 3} \quad \text{for } x \in \mathbb{R}, x \neq 3.$$

1. Obtain expressions, in terms of \(x\), for \(f^{-1}(x)\) and \(g^{-1}(x)\), stating the value of \(x\) for which \(g^{-1}(x)\) is not defined. [4]
2. Sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) on the same diagram, making clear the relationship between the two graphs. [3]
3. Given that the equation \(fg(x) = 5 - kx\), where \(k\) is a constant, has no solutions, find the set of possible values of \(k\). [5]

The function \(f\) is such that \(f(x) = 8 - (x - 2)^2\), for \(x \in \mathbb{R}\).

1. Find the coordinates and the nature of the stationary point on the curve \(y = f(x)\). [3]

The function \(g\) is such that \(g(x) = 8 - (x - 2)^2\), for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.

1. State the smallest value of \(k\) for which \(g\) has an inverse. [1]

For this value of \(k\),

1. find an expression for \(g^{-1}(x)\), [3]
2. sketch, on the same diagram, the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\). [3]

Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by \begin{align} \mathrm{f} : x \mapsto 2x + 3 \quad &\text{for } x \leqslant 0,
\mathrm{g} : x \mapsto x^2 - 6x \quad &\text{for } x \leqslant 3. \end{align}

1. Express \(\mathrm{f}^{-1}(x)\) in terms of \(x\) and solve the equation \(\mathrm{f}(x) = \mathrm{f}^{-1}(x)\). [3]
2. On the same diagram sketch the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\), showing the coordinates of their point of intersection and the relationship between the graphs. [3]
3. Find the set of values of \(x\) which satisfy \(\mathrm{gf}(x) \leqslant 16\). [5]

The function \(f : x \mapsto 6 - 4\cos(\frac{1}{2}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).

1. Find the exact value of \(x\) for which \(f(x) = 4\). [3]
2. State the range of \(f\). [2]
3. Sketch the graph of \(y = f(x)\). [2]
4. Find an expression for \(f^{-1}(x)\). [3]

The curve \(C\) has equation \(y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}\).

1. Find the equations of the asymptotes of \(C\). [2]
2. Find the coordinates of any stationary points on \(C\). [4]
3. Sketch \(C\), stating the coordinates of any intersections with the axes. [5]
4. Sketch the curve with equation \(y = \left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right|\) and state the set of values of \(k\) for which \(\left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right| = k\) has 4 distinct real solutions. [2]

1. Sketch, on the same set of axes, the graphs of $$y = 2 - e^{-x} \text{ and } y = \sqrt{x}.$$ [3]

[It is not necessary to find the coordinates of any points of intersection with the axes.] Given that \(f(x) = e^{-x} + \sqrt{x} - 2, x \geq 0\),

1. explain how your graphs show that the equation \(f(x) = 0\) has only one solution, [1]
2. show that the solution of \(f(x) = 0\) lies between \(x = 3\) and \(x = 4\). [2]

The iterative formula \(x_{n+1} = (2 - e^{-x_n})^2\) is used to solve the equation \(f(x) = 0\).

1. Taking \(x_0 = 4\), write down the values of \(x_1, x_2, x_3\) and \(x_4\), and hence find an approximation to the solution of \(f(x) = 0\), giving your answer to 3 decimal places. [4]

28a.

1. Given that \(\cos(x + 30)° = 3 \cos(x - 30)°\), prove that \(\tan x° = -\frac{\sqrt{3}}{2}\). [5]
2. 1. Prove that \(\frac{1 - \cos 2\theta}{\sin 2\theta} = \tan \theta\). [3]
   2. Verify that \(\theta = 180°\) is a solution of the equation \(\sin 2\theta = 2 - 2 \cos 2\theta\). [1]
   3. Using the result in part (a), or otherwise, find the other two solutions, \(0 < \theta < 360°\), of the equation using \(\sin 2\theta = 2 - 2 \cos 2\theta\). [4]

The function \(f\) is even and has domain \(\mathbb{R}\). For \(x \geq 0\), \(f(x) = x^2 - 4ax\), where \(a\) is a positive constant.

1. In the space below, sketch the curve with equation \(y = f(x)\), showing the coordinates of all the points at which the curve meets the axes. [3]
2. Find, in terms of \(a\), the value of \(f(2a)\) and the value of \(f(-2a)\). [2]

Given that \(a = 3\),

1. use algebra to find the values of \(x\) for which \(f(x) = 45\). [4]

Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\).

1. On the insert, on the same axes, plot the graph of \(y = x^2 - 5x + 5\) for \(0 \leq x \leq 5\). [4]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac{1}{x}\) and \(y = x^2 - 5x + 5\) satisfy the equation \(x^3 - 5x^2 + 5x - 1 = 0\). [2]
3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x^3 - 5x^2 + 5x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x^3 - 5x^2 + 5x - 1 = 0\) is rational. [5]

$$f(x) = x - [x], \quad x \geq 0$$ where \([x]\) is the largest integer \(\leq x\). For example, \(f(3.7) = 3.7 - 3 = 0.7\); \(f(3) = 3 - 3 = 0\).

1. Sketch the graph of \(y = f(x)\) for \(0 \leq x < 4\). [3]
2. Find the value of \(p\) for which \(\int_2^p f(x) dx = 0.18\). [3]

Given that $$g(x) = \frac{1}{1+kx}, \quad x \geq 0, \quad k > 0,$$ and that \(x_0 = \frac{1}{2}\) is a root of the equation \(f(x) = g(x)\),

1. find the value of \(k\). [2]
2. Add a sketch of the graph of \(y = g(x)\) to your answer to part \((a)\). [1]

The root of \(f(x) = g(x)\) in the interval \(n < x < n + 1\) is \(x_n\), where \(n\) is an integer.

1. Prove that $$2 x_n^2 - (2n - 1)x_n - (n + 1) = 0.$$ [4]
2. Find the smallest value of \(n\) for which \(x_n - n < 0.05\). [4]

A curve has equation $$y = \frac{5 - 4x}{1 + x}$$

1. Sketch the curve. [4 marks]
2. Hence sketch the graph of \(y = \left|\frac{5 - 4x}{1 + x}\right|\). [1 mark]

Functions f, g and h are defined for \(x \in \mathbb{R}\) by $$f : x \mapsto x^2 - 2x,$$ $$g : x \mapsto x^2,$$ $$h : x \mapsto \sin x.$$

1. 1. State whether or not f has an inverse, giving a reason. [2]
   2. Determine the range of the function f. [2]
2. 1. Show that gh(x) can be expressed as \(\frac{1}{2}(1 - \cos 2x)\). [2]
   2. Sketch the curve C defined by \(y = \text{gh}(x)\) for \(0 \leqslant x \leqslant 2\pi\). [3]

% Figure shows a curve with maximum at point A, passing through origin O, with horizontal asymptote \includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\) where $$f(x) = \frac{x^2 + 16}{3x} \quad x \neq 0$$ The curve has a maximum at the point \(A\) with coordinates \((a, b)\).

1. Find the value of \(a\) and the value of \(b\). [4] The function g is defined as $$g : x \mapsto \frac{x^2 + 16}{3x} \quad a \leq x < 0$$ where \(a\) is the value found in part (a).
2. Write down the range of g. [1]
3. On the same axes sketch \(y = g(x)\) and \(y = g^{-1}(x)\). [3]
4. Find an expression for \(g^{-1}(x)\) and state the domain of \(g^{-1}\) [5]
5. Solve the equation \(g(x) = g^{-1}(x)\). [3]