Exam 1: Functions and Their Graphs

Hooke's Law states that the force $F$ required to compress or stretch a spring (within its elastic limits) is proportional to the distance $d$ that the spring is compressed or stretched from its original length. That is, $F = k d$ , where $k$ is the measure of the stiffness of the spring and is called the spring constant. The table below shows the elongation $d$ in centimeters of a spring when a force of $F$ kilograms is applied. Force, Elongation, 20 1.4 40 2.5 60 4.0 80 5.3 100 6.6 Find the equation of the line that seems to best fit the data. Use the model to estimate the elongation of the spring when a force of 55 kilograms is applied. Round your answer to one decimal place.

Use function notation to write $g$ in terms of $f ( x ) = x ^ { 3 }$ . $g ( x ) = - \frac { 1 } { 4 } ( x + 9 ) ^ { 3 }$

Evaluate the function at the specified value of the independent variable and simplify. $f ( y ) = 2 y + 7$ $f ( - 1.4 )$

Estimate the slope of the line.

If $f$ is an even function, determine if $g$ is even, odd, or neither. $g ( x ) = f ( - x ) + 1$

Determine an equation that may represented by the graph shown below.

Hooke's Law states that the force $F$ required to compress or stretch a spring (within its elastic limits) is proportional to the distance $d$ that the spring is compressed or stretched from its original length. That is, $F = k d$ , where $k$ is the measure of the stiffness of the spring and is called the spring constant. The table below shows the elongation $d$ in centimeters of a spring when a force of $F$ kilograms is applied. Force, Elongation, 20 2.8 40 5.0 60 8.0 80 10.6 100 13.2 Find the equation of the line that seems to best fit the data. Use the model to estimate the elongation of the spring when a force of 50 kilograms is applied. Round your answer to one decimal place.

Estimate the slope of the line.

Algebraically determine whether the function below is even, odd, or neither. $f ( s ) = 8 s ^ { 7 / 6 }$

Find the inverse function of $f$ . $f ( x ) = x ^ { 9 } - 2$

Does the table describe a function? Input value 5 10 13 10 5 Output value -13 -9 0 9 13

The scatter plots of different data are shown below. Determine whether there is a positive correlation, negative correlation, or no discernible correlation between the variables.

Use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant. $f ( x ) = 2 x ^ { 4 } - 4 x ^ { 2 }$

Find the domain of the function. $g ( x ) = \sqrt { 25 - x ^ { 2 } }$

Find the difference quotient and simplify your answer. $f ( w ) = - 9 w ^ { 2 } + 2 w , \quad \frac { f ( 4 + h ) - f ( 4 ) } { h } , h \neq 0$

Which equation does not represent $y$ as a function of $x$ ?

Write the slope-intercept form of the equation of the line through the given point parallel to the given line. point: $( 3 , - 4 )$ line: $28 x + 7 y = - 4$

Find the value(s) of $x$ for which $f ( x ) = g ( x )$ . $f ( x ) = x ^ { 2 } - 7 x + 3 \quad g ( x ) = - 3 x + 8$

Determine the domain and range of the inverse function $f ^ { - 1 }$ of the following function $f$ . $f ( x ) = - | x + 8 | - 3$ , where $x > - 8$

Find the domain of the function. $g ( w ) = \frac { - 7 w } { w - 5 }$