Q 65. - Precalculus Enhanced with Graphing Utilities

Question

(a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing.

$g\left(x\right)=-2{\left(x-3\right)}^2+2$

Step-by-Step Solution

Answer

Part (a). The required graph is:

Part (b). The domain of the function is all real numbers and the range of the function is $(-\infty ,2]$.

Part (c). The function is increasing in $\left(-\infty ,3\right)$ and decreasing in $\left(3,\infty \right)$.

Part (a). Step 1. Given information.

The given function is $g\left(x\right)=-2{\left(x-3\right)}^2+2$.

Part (a). Step 2. Draw the graph.

Simplify the given function:

$\begin{array}{rcl}g\left(x\right)& =& -2{\left(x-3\right)}^2+2\\ & =& -2\left(x^2-6x+9\right)+2\\ & =& -2x^2+12x-18+2\\ & =& -2x^2+12x-16\end{array}$

The required graph is:

Part (b). Step 1. Given information.

The given function is $g\left(x\right)=-2{\left(x-3\right)}^2+2$.

Part (b). Step 2. Determine the domain and range of the given function.

From the above graph, the domain of the function is $\left(-\infty ,+\infty \right)$ and the range of the function is $(-\infty ,2]$.

Part (c). Step 1. Given information.

The given function is $g\left(x\right)=-2{\left(x-3\right)}^2+2$.

Part (c). Step 2. Determine where the function is decreasing and increasing.

The function is increasing in $\left(-\infty ,3\right)$ and decreasing in $\left(3,\infty \right)$.