I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 1.1 : Review : Functions
19. Find the domain and range of \(g\left( z \right) = - {z^2} - 4z + 7\).
Show SolutionThis 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,
\[{\mbox{Range : }}\,\left( { - \infty ,11} \right]\]