OCR MEI C3 (Core Mathematics 3)

1. Show algebraically that the function \(\text{f}(x) = \frac{2x}{1-x^2}\) is odd. [2] Fig. 7 shows the curve \(y = \text{f}(x)\) for \(0 \leq x < 4\), together with the asymptote \(x = 1\). \includegraphics{figure_7}
2. Use the copy of Fig. 7 to complete the curve for \(-4 \leq x \leq 4\). [2]

The functions f(x) and g(x) are defined as follows. $$\text{f}(x) = \ln x, \quad x > 0$$ $$\text{g}(x) = 1 + x^2, \quad x \in \mathbb{R}$$ Write down the functions fg(x) and gf(x), and state whether these functions are odd, even or neither. [4]

Each of the graphs of \(y = \text{f}(x)\) and \(y = \text{g}(x)\) below is obtained using a sequence of two transformations applied to the corresponding dashed graph. In each case, state suitable transformations, and hence find expressions for f(x) and g(x).

1. \includegraphics{figure_3i} [3]
2. \includegraphics{figure_3ii} [3]

Fig. 4 shows the curve \(y = \text{f}(x)\), where \(\text{f}(x) = \sqrt{1 - 9x^2}\), \(-a < x < a\). \includegraphics{figure_4}

1. Find the value of \(a\). [2]
2. Write down the range of f(x). [1]
3. Sketch the curve \(y = \text{f}(\frac{1}{3}x) - 1\). [3]

You are given that f(x) and g(x) are odd functions, defined for \(x \in \mathbb{R}\).

1. Given that s(x) = f(x) + g(x), prove that s(x) is an odd function. [2]
2. Given that p(x) = f(x)g(x), determine whether p(x) is odd, even or neither. [2]

1. State the algebraic condition for the function f(x) to be an even function. What geometrical property does the graph of an even function have? [2]
2. State whether the following functions are odd, even or neither. (A) \(\text{f}(x) = x^2 - 3\) (B) \(\text{g}(x) = \sin x + \cos x\) (C) \(\text{h}(x) = \frac{1}{x + x^3}\) [3]

Fig. 8 shows part of the curve \(y = \text{f}(x)\), where \(\text{f}(x) = e^{-\frac{1}{5}x} \sin x\), for all \(x\). \includegraphics{figure_8}

1. Sketch the graphs of (A) \(y = \text{f}(2x)\), (B) \(y = \text{f}(x + \pi)\). [4]
2. Show that the \(x\)-coordinate of the turning point P satisfies the equation \(\tan x = 5\). Hence find the coordinates of P. [6]
3. Show that \(\text{f}(x + \pi) = -e^{-\frac{1}{5}\pi}\text{f}(x)\). Hence, using the substitution \(u = x - \pi\), show that $$\int_{\pi}^{2\pi} \text{f}(x)\,dx = -e^{-\frac{1}{5}\pi} \int_{0}^{\pi} \text{f}(u)\,du.$$ Interpret this result graphically. [You should not attempt to integrate f(x).] [8]