1.02u Functions: definition and vocabulary (domain, range, mapping)

In this question you must show detailed reasoning. \includegraphics{figure_5} The function f is defined for the domain \(x \geqslant 0\) by $$\mathrm{f}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ The diagram shows the curve \(y = \mathrm{f}(x)\).

1. Find the range of f. [6]

1. The function g is defined for the domain \(x \geqslant k\) by $$\mathrm{g}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ Given that g is a one-one function, state the least possible value of \(k\). [1]

1. Find the exact area of the shaded region enclosed by the curve and the \(x\)-axis. [7]

For all real values of \(x\), the functions f and g are defined by \(f(x) = x^2 + 8ax + 4a^2\) and \(g(x) = 6x - 2a\), where \(a\) is a positive constant.

1. Find fg\((x)\). Determine the range of fg\((x)\) in terms of \(a\). [4]
2. If fg\((2) = 144\), find the value of \(a\). [3]
3. Determine whether the function fg has an inverse. [2]

Let \(f(x) = x^2(x - 2)\) and \(g(x) = 2x - 1\) for all real \(x\).

1. Sketch the graph of \(y = f(x)\) and explain briefly why the function f has no inverse. [2]
2. Write down \(g^{-1}(x)\). [1]
3. On the same diagram, sketch the graphs of \(y = f(x - 1) - 3\) and \(y = g^{-1}(x)\) and state the number of real roots of the equation \(f(x - 1) - 3 = g^{-1}(x)\). [3]

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]