Exam 1: Functions and Their Graphs

A rectangle is bounded by the x-axis and the semicircle $y = \sqrt { 49 - x ^ { 2 } }$ (see figure).Select the area A of the rectangle as a function of x, and determine the domain of the function.

Find ( f / g )(x). $f ( x ) = x ^ { 2 } - 4 x \quad g ( x ) = 7 - x$

Determine whether the function has an inverse function.If it does, find the inverse function. $h ( x ) = - \frac { 4 } { x ^ { 2 } }$

The frequency of vibrations of a piano string varies directly as the square root of the tension on the string and inversely as the length of the string.The middle A string has a frequency of 430 vibrations per second.Find the frequency of a string that has 1.25 times as much tension and is 1.4 times as long. ​

A force of $f = 225$ newtons stretches a spring $s = 0.15$ meter (see figure). How far will a force of 60 newtons stretch the spring? What force is required to stretch the spring 0.1 meter

Your wage is $11.00 per hour plus $0.50 for each unit produced per hour.So, your hourly wage in terms of the number of units produced x is y = 11 + 0.50x.Find the inverse function.What does each variable represent in the inverse function

Determine whether the lines are parallel, perpendicular, or neither. L1: y = $\frac { 1 } { 2 }$ x - 4 L2: y = - $\frac { 1 } { 2 }$ x - 2

Plot the following points and find the midpoint of the line segment joining the points. ​ (9, 6), (5, 6) ​

The number N of bacteria in a refrigerated food is given by $N ( T ) = 10 T ^ { 2 } - 20 T + 600,1 \leq T \leq 20$ where T is the temperature of the food in degrees Celsius.When the food is removed from refrigeration, the temperature of the food is given by $T ( t ) = 3 t + 2,0 \leq t \leq 6$ where t is the time in hours. Find the bacteria count after 0.5 hour.

Draw the line using the slope and y-intercept. ​ Y + 1 = 2x ​

The number of lumens (time rate of flow of light) L from a fluorescent lamp can be approximated by the model ​ L = -0.294x² + 97.744x - 664.875, 20 ≤ x ≤ 90 ​ Where x is the wattage of the lamp. ​ Use a graphing utility to select the graph of the function.Use the graph to estimate the wattage necessary to obtain 2400 lumens. ​

Find $g \circ f$ and the domain of the composite function. $f ( x ) = \frac { 1 } { x } , g ( x ) = x + 5$

Estimate the slope of the line. ​ ​

Find the slope and y-intercept (if possible) of the equation of the line.Select the correct answer for the line. $y = 8 x + 1$

Find a mathematical model representing the statement.(Determine the constant of proportionality.) Z varies directly as the square of x and inversely as y.(z = 36 when x = 9 and y = 3.)

Determine whether the function has an inverse function.If it does, find the inverse function. $g ( x ) = \frac { x } { 5 }$

Find $f \circ g$ . $f ( x ) = x + 3 \quad g ( x ) = \frac { 1 } { x ^ { 2 } - 9 }$

Determine which of the following point lies on the graph of the equation. $x ^ { 2 } + y ^ { 2 } = 5$

Write an equation for the function that is described by the given characteristic. The shape of $f ( x ) = x ^ { 3 }$ , but shifted 17 units to the right.

Use the intercept form to find the equation of the line with the given intercepts.The intercept form of the equation of a line with intercepts (a, 0) and (0, b) is $\frac { x } { a } + \frac { y } { b } = 1 , a \neq 0 , b \neq 0$ X-intercept: ( $- \frac { 5 } { 8 }$ , 0) Y-intercept: $( 0 , - 3 )$