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Section 1.1 : Review : Functions

19. Find the domain and range of \(g\left( z \right) = - {z^2} - 4z + 7\).

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This is a polynomial (a 2nd degree polynomial in fact) and so we know that we can plug any value of \(z\) into the function and so the domain is all real numbers or,

\[{\mbox{Domain : }} - \infty < z < \infty \,\,\,{\rm{or}}\,\,\,\left( { - \infty ,\infty } \right)\]

The graph of this 2nd degree polynomial (or quadratic) is a parabola that opens downwards (because the coefficient of the \({z^2}\) is negative) and so we know that the vertex will be the highest point on the graph. This also means that the function will take on all values less than or equal to the \(y\)-coordinate of the vertex which will in turn give us the range.

So, we need the vertex of the parabola. The \(z\)-coordinate is,

\[z = - \frac{{ - 4}}{{2\left( { - 1} \right)}} = - 2\]

and the \(y\) coordinate is then, \(g\left( { - 2} \right) = 11\).

The range is then,

\[{\mbox{Range : }}\,\left( { - \infty ,11} \right]\]