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The diagram shows the graphs of three functions, \(f _ { 1 } , f _ { 2 }\) and \(f _ { 3 }\). The function \(f _ { 1 }\) is a probability density function.
State the value of \(k\).
For each of the functions \(\mathrm { f } _ { 2 }\) and \(\mathrm { f } _ { 3 }\), state why it cannot be a probability density function.
The probability density function g is defined by
$$g ( x ) = \begin{cases} 6 \left( a ^ { 2 } - x ^ { 2 } \right) & - a \leqslant x \leqslant a 0 & \text { otherwise } \end{cases}$$
where \(a\) is a constant.