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Section 1.1 : Review : Functions

21. Find the domain and range of \(h\left( y \right) = - 3\sqrt {14 + 3y} \).

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In this case we need to require that,

\[14 + 3y \ge 0\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}y \ge - \frac{{14}}{3}\]

in order to make sure that we don’t take the square root of negative numbers. The domain is then,

\[{\mbox{Domain : }} - \frac{{14}}{3} \le y < \infty \,\,\,{\rm{or}}\,\,\,\left[ { - \frac{{14}}{3},\infty } \right)\]

For the range for this function we can notice that the quantity under the root can be zero (if \(y = - \frac{14}{3}\)). Also note that because the quantity under the root is a line it will take on all positive values and so the square root will in turn take on all positive value and zero. The square root is then multiplied by -3. This won’t change the fact that the root can be zero, but the minus sign will change the sign of the non-zero values from positive to negative. The 3 will only affect the general size of the square root but it won’t change the fact that the square root will still take on all positive (or negative after we add in the minus sign) values.

The range is then,

\[{\mbox{Range : }}\left( { - \infty ,0} \right]\]