Exam 3: Polynomial and Rational Functions

Describe how the graph of $y = - f ( 2 x ) + 2$ can be obtained from the graph of $f .$

A function f is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. $f ( x ) = | x |$ ; shift to the left $1 / 2$ unit, shrink vertically by a factor of 0.2, and shift downward 2 units.

Find the range of the function. $f ( x ) = \sqrt { x + 4 }$

$\text { Evaluate } f ( - 2 ) , f ( - 1 ) , f ( 0 ) , f ( 1 ) \text {, and } f ( 5 ) \text { for the piecevise-defined function }$ $f ( x ) = \left\{ \begin{array} { l l } 3 x ^ { 2 } & \text { if } x < 0 \\2 x + 1 & \text { if } x \geq 0\end{array} \right.$

Determine which functions are one-to-one. I II III

Evaluate $f ( - 1 )$ , $f ( 0 )$ , $f ( 1 )$ , for the piecewise-defined function. $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & \text { if } x < 0 \\2 x - 1 & \text { if } x \geq 0\end{array} \right.$

Determine if the equation $x ^ { 2 } + y ^ { 2 } - 25 = 0$ defines $y$ as a function of $x$ . Explain your answer.

Given $f ( x ) = 2 + x ^ { 2 }$ and $g ( x ) = \sqrt { x - 1 }$ , find $( f \circ g ) ( 2 )$ , $( f \circ f ) ( 2 )$

The graph shows the depth of water W in a reservoir over a one-year period as a function of the number of days x since the beginning of the year. Estimate the average rate of change of W between x = 0 and x= 100?

A function f is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. $f ( x ) = | x |$ ; shift to the left $\frac { 1 } { 2 }$ unit, shrink vertically by a factor of 0.2, and shift downward 2 units.

For the function given, find $g ( - 1 )$ , $g ( 3 )$ , and $g \left( a ^ { 2 } \right)$ $g ( x ) = ( 1 / x ) + x ^ { 2 }$

Use a graphing device to draw the graph of the function $f ( x ) = 144 x ^ { 3 } - 144 x ^ { 2 } + 36 x$ State approximately the intervals on which the function is increasing and on which the function is decreasing.

Find the domain of the function. $f ( x ) = \frac { x + 3 } { x ^ { 2 } - 4 }$

Determine whether the curve represents a graph of a function.

Find the inverse of the function. $f ( x ) = \sqrt { 25 - x ^ { 2 } }$ , $0 \leq x \leq 5$

Given $f ( x ) = \frac { 1 } { x + 2 }$ and $g ( x ) = \frac { 1 } { x - 2 }$ , find $f \circ g$ , $g \circ f$ .

Use a graphing device to draw the graph of the function $f ( x ) = - 3 - 3 x ^ { 2 }$ . State approximately the intervals on which the function is increasing and on which the function is decreasing.

The graph shows the depth of water W in a reservoir over a one-year period as a function of the number of days x since the beginning of the year. What was the average rate of change of W between x = 0 and x= 100?

Use a graphing calculator or computer to determine whether or not the function $f ( x ) = - | x | - | 5 - x |$ is one-to-one.

Find the range of the function. $f ( x ) = 2 x ^ { 2 } - 3$