Exam 3: Polynomial and Rational Functions

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = x ^ { 2 } , \quad - 3 \leq x \leq 5$

Find the inverse of the function. $f ( x ) = \frac { x - 2 } { 3 }$

If $f ( x ) = x ^ { 3 } + 2 x - 1$ , find $f ( a )$

Use a graphing calculator to determine approximately the intervals on which the function is increasing, and on which $f$ is decreasing. $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } + x ^ { 2 } - 24 x + 16$

Determine whether each curve represents a graph of a function. a.) b.) c.) d.)

If $f ( x ) = 3 x - 2$ and $g ( x ) = 3 + 2 x ^ { 2 }$ , find $f g$ and $( f \circ g ) ( x )$

A function f is given, and the indicated transformations are applied to its graph (in the given order). Find the equation for the final transformed graph. $f ( x ) = \sqrt { x }$ ; shift 5 units to the left, stretch vertically by a factor of 2, and reflect in the x-axis

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

Determine whether or not the function $f ( x ) = x ^ { 2 } - 3 x + 2$ is one-to-one.

For the function $g ( t ) = \frac { 1 } { 3 t - 2 }$ determine the average rate of change between the values $t = 0$ and $t = a + 1$ .

The graph gives the number of farms in the United States from 1850 to 2000.Estimate the average rate of change in the number of farms between the following years. (i) 1860 and 1890 (ii) 1920 and 1980

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

The graph of a function $h$ is given. (a) Find $h ( - 3 ) , h ( - 2 ) , h ( 2 )$ , and $h ( 4 )$ (b) Find the domain and range of $h$ . (c) Find the values of $x$ for which $h ( x ) = 3$ (d) Find the values of $x$ for which $h ( x ) \leq 3$ .

Describe how the graph of $y = - f ( 3 x ) + 4$ can be obtained from the graph of $f .$

Determine which viewing rectangle produces the most appropriate graph of the function. $g ( x ) = 6 x ^ { 3 } - 15 x ^ { 2 } + 4 x - 1$

Use $f ( x ) = 3 x - 2$ and $g ( x ) = 3 + 2 x ^ { 2 }$ to evaluate the expression $( f \circ g ) ( 2 )$

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

For the function $f ( x ) = 2 x ^ { 2 } - x$ determine the average rate of change between the values $x = - 1$ and $x = 0$

Sketch the graph of the function. $G ( x ) = x ^ { 3 } - 3 x ^ { 2 }$

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