- Take the function \(f(x)= \sqrt{x-3}\). Its domain is \(x \geq 3\), and its range is \(f(x) \geq 0\). Which of the following is true of \(f\)?
- If \(x \geq 3\), then \(f(x) \geq 0\).
- It assigns exactly one value to every positive value of \(x\).
- The range of the function is \(f(3)\).
- The value of \(f(3)\) is undefined.
The function \(f\) assigns to each element of the domain exactly one element of the range. Consequently, if \(x\) is in the domain \(x \geq 3\), then the value of \(f\) is in the range \(f(x) \geq 0\).
- If \(g(x) = 3x + x + 5\), evaluate \(g(2)\).
- \(g(2)=8\)
- \(g(2)=9\)
- \(g(2)=13\)
- \(g(2)=17\)
To evaluate \(g(2)\), substitute 2 for every occurrence of \(x\) in the equation. Then, simplify the result using order of operations:
\(g(2) = 3(2) + (2) + 5\)
\(= 6 + 2 + 5\)
\(= 13\)
- The function \(S(r) =4\pi r^2\) gives the surface area of a sphere of radius \(r\). What is the surface area of a sphere of radius 4?
- \(8 \pi\)
- \(16 \pi\)
- \(32 \pi\)
- \(64 \pi\)
The surface area will be given by the expression \(S(4)\). To calculate this value, substitute 4 for \(r\) in the equation \(S(r)=4\pi r^2\). Then, simplify the result using order of operations:
\(S(4)=4\pi(4)^2\)
\(= 4\pi \cdot 16\)
\(= 64 \pi\)
- A theater will sell 500 tickets to a play if it charges $10 per ticket. Furthermore, every time it raises the price by $1, it will sell 50 fewer tickets. Which of the following functions represents the number of tickets the theater will sell if it charges \(d\) dollars per ticket?
- \(t(d) = -50d + 500\)
- \(t(d) = -50d + 1,000\)
- \(t(d) = 50d + 10\)
- \(t(d) = 50d + 100\)
We are given that the theater sells 500 tickets when the price is $10. Additionally, each $1 increase in price results in 50 fewer tickets being sold. This indicates a linear relationship with a negative slope.
The number of tickets sold decreases by 50 for every $1 increase in ticket price, so the slope is –50. The general form of the function is:
\(t(d) = -50d + c\)
We’re told that when \(d = 10\), \(t(d) = 500\). Substitute into the equation to solve for \(c\):
\(500 = -50(10) + c\)
\(500 = -500 + c\)
\(c = 1,000\)
- A taxi ride costs $4.25 for the first mile and $0.70 for each mile after the first. Which of the following functions gives the total cost (in dollars) of traveling \(d\) miles (assuming that \(d \geq 1\))?
- \(c(d) = 3.55 + 0.70d\)
- \(c(d) = 3.55 + 0.70(d-1)\)
- \(c(d) = 4.25 + 0.70d\)
- \(c(d) = 4.25 + 0.70(d-1)\)
The cost of the taxi ride is the sum of two functions, a constant function for the first mile and a linear function for the rest of the ride.
The constant function is \(c_1 (d)=4.25\) since the cost of the first mile is $4.25. For the linear part, subtract 1 from \(d\) to exclude the first mile, and then multiply the result by 0.70 since it costs $0.70 per mile.
The result is \(c_2 (d)=0.70(d-1)\). Finally, write the function for the total cost of the taxi ride by adding the two functions.
\(c(d)=c_1 (d)+c_2 (d) \)
\(= 4.25 + 0.70(d-1)\)
- Exponential functions grow by equal factors over equal intervals. By what factor does the exponential function \(f(x)=3\cdot 2^x\) grow by over every interval whose length is 3?
- By a factor of 6
- By a factor of 8
- By a factor of 18
- By a factor of 24
The length of an interval is the difference between its endpoints. For example, the length of the interval \([2,4]\) is 2.
To determine how the given function grows over an interval of length 3, determine the value of \(f\) at each endpoint of that interval. Since exponential functions grow by equal factors over equal intervals, you can use any interval of length 3, and your answer will apply to all such intervals.
For example, you can use the interval \([0,3]\):
\(f(0) = 3 \cdot 2^0\)
\(= 3 \cdot 1\)
\(= 3\)
\(f(3)=3 \cdot 2^3\)
\(= 3 \cdot 8\)
\(= 24\)
- A linear function can be used to convert a temperature from Fahrenheit to Celsius. For example, you can use it to convert 32°F to 0°C and 68°F to 20°C. Use this information to convert 104°F to Celsius.
- 25°C
- 30°C
- 40°C
- 50°C
Linear functions grow by equal differences (rather than equal factors) over equal intervals. In other words, if the linear function \(c(f)\) converts a temperature \(f\) from Fahrenheit to Celsius, then intervals of equal length (in \(f\)) result in equal increases in the value of the function \(c(f)\).
From the problem, we know that \(c(32) = 0\) and \(c(68) = 20\). Thus, we can conclude that intervals of length 36 (like the interval \([32,68]\)) result in an increase of 20.
In addition, since the length of \([68,104]\) is 36, the function \(c(f)\) increases by 20 over this interval as well. Use this information to calculate \(c(104)\).
\(c(104) = c(68) + 20 = 20 + 20 = 40\)
- In the figure below, circle O is a unit circle, and the measure of \(\angle AOB\) is \(\tfrac{\pi}{3}\). What is the length of \(\widehat{AB}\) ?
- \(\tfrac{\pi}{6}\)
- \(\tfrac{\pi}{3}\)
- \(\tfrac{2\pi}{3}\)
- \(\pi\)
An arc is a piece of a circle. In the figure, \(AB\) is the piece of the circle that starts at point A and ends at B.
In general, an arc length \(s\) is given by \(s = \theta R\), where \(R\) is the radius of the circle containing the arc and \(\theta\) is the angle subtended by radii drawn to the endpoints of the arc.
In a unit circle, the length of an arc is simply the measure of the angle (in radians) subtended by the angle. Therefore, the arc length of \(AB\) is equal to the measure of \(\angle AOB\), so its length is \(\tfrac{\pi}{3}\).
- A unit circle and an angle are graphed on the coordinate plane below. Use the graph to calculate the value of \(\text{sin } \theta\) .
- \(\text{sin } \theta = 0.8\)
- \(\text{sin } \theta = 0.6\)
- \(\text{sin } \theta = -0.6\)
- \(\text{sin } \theta = -0.8\)
In the unit circle, the sine of an angle is equal to the \(y\)-coordinate of the point where the terminal side of the angle intersects the circle. Angles are measured from the positive \(x\)-axis, moving counterclockwise.
In this graph, the terminal side of the angle \(\theta\) lands in quadrant III, where both \(x\)– and \(y\)-coordinates are negative. The point where it intersects the unit circle is approximately \((–0.8,–0.6)\).
Since the sine of the angle is the \(y\)-coordinate:
\(\text{sin }\theta=−0.6\)
So, the correct value of \(\text{sin }\theta\) is −0.6.
- Calculate the value of \(\text{tan }(\tfrac{\pi}{6})\) .
- \(\text{tan }(\tfrac{\pi}{6})=\tfrac{1}{2}\)
- \(\text{tan }(\tfrac{\pi}{6})=\tfrac{\sqrt{3}}{3}\)
- \(\text{tan }(\tfrac{\pi}{6})=\sqrt{3}\)
- \(\text{tan }(\tfrac{\pi}{6})=2\)
The angle \(\tfrac{\pi}{6}\) is in radians. To convert it to degrees, multiply by \(\tfrac{180}{\pi}\).
\(\dfrac{\pi}{6}\cdot \dfrac{180}{\pi}=30°\)
Therefore, \(\text{tan } \tfrac{\pi}{6}=\text{tan }30°\).
To calculate this value, draw a 30-60-90 triangle, which is a special triangle whose proportions you may have memorized. Making the hypotenuse one unit long simplifies things, although it is not necessary as long as the proportions are the same.
By SOHCAHTOA, the tangent function is defined as \(\tfrac{\text{opposite}}{\text{adjacent}}\) in a right triangle. Therefore, the value of \(\text{tan } 30°\) is \(\tfrac{\tfrac{1}{2}}{\sqrt{\tfrac{3}{2}}}\).
Simplify this fraction:
\(\text{tan } \tfrac{\pi}{6}=\tfrac{\tfrac{1}{2}}{\sqrt{\tfrac{3}{2}}}\)
\(= \tfrac{1}{\sqrt{3}}\)
\(=\tfrac{\sqrt{3}}{3}\)