Section 1.4 Projects: Functions and Notation
¶Exercises 1.4.1 Exercises
Subsection 1.4.1.1 Evaluate, apply and interpret function notation, including the notation for inverse functions and composition of Functions
1.
The graph in the figure models the amount of drug, \(A(t)\) mg, in a patient’s body after ingestion as a function of time, \(t\) hours. The body absorbs the drug at a linear rate, but then eliminates the drug at an exponential rate
Estimate the interval over which \(A(t)\) is decreasing.
Evaluate \(A(10)\text{.}\) Use units to interpret the meaning of your answer in the context of the situation.
Solve \(A(t) = 100\text{.}\) Use units to interpret the meaning of your answer in the context of the situation.
Calculate on the interval \(\frac{\Delta A}{\Delta t}\) on the interval \(3 \leq x \leq 6\text{.}\) Show your work.
What does your answer to part \((d)\) mean in terms of the situation? Use units to interpret the meaning of your answer in the context of the situation.
2.
The function W(h) represents the average weight (measured in pounds) of 60 year old American men as a function of their height (measured in centimeters). Suppose \(W^{1}(181)=180\text{.}\) Explain what this means in a complete sentence.
3.
The function \(A(r)=2 \pi r^2+20 \pi r\) represents the surface area (measured in square inches) of a cylinder that is \(10\) inches tall with the radius of the circular base equal to \(r\) inches.
Explain what \(A^{1}(150)\) represents. Be specific.
What does \(A(3)\) represent?
What is a likely domain for the function \(A(r)\text{?}\)
Is the function \(A(r)\) increasing or decreasing?

Match each expression to the correct phrase:
\(A(3r)\) The surface area of three cans of radius \(r\) \(3A(r)\) The surface area of a can of radius \(r\text{,}\) then add on \(3\) more square inches \(A(r)+3\) The surface area of a can whose radius is \(3\) times as large as \(r\) Table 1.4.4.
4.
The graph in the figure shows two functions \(f(x)\text{,}\) linear, and \(g(x)\text{,}\) quadratic.
Sketch the graph of \(f(x)g(x)\text{.}\) Clearly label at least 6 different points on the resulting graph, and connect them with a smooth curve.
Let \(S(x)=f(x)+g(x)\text{,}\) \(P(x)=f(x)*g(x)\) and \(Q(x)= \frac{f(x)}{g(x)}\)
Evaluate \(S(2)\)
Evaluate \(P(8)\)
Solve \(Q(x)=0\)
5.
A manufacturer purchases raw materials from a supplier and then makes products to sell. The amount of money the manufacturer makes from selling one of these products depends on how much they have spend on the materials. At the same time, however, the price of materials changes throughout the year.
Exstimate \(f(35)\) and interpret its meaning.
At \(40\) days after January \(1^{\rm{st}}\text{,}\) how much will they sell the product for?
Evaluate \(f(P(120))\) and interpret its meaning.
What is wrong with the composition \(P(f(20))\text{?}\) [Hint: Think about the units involved in the expression.]
Use a composition to write an equation that means: “If the company buys their materials \(280\) days after January \(1^{\rm{st}}\text{,}\) they will sell the product for \(\$ 1.19\text{.}\)”
Explain the meaning of \(f(P(0))=2\text{.}\)
Solve the equation \(f(P(t))=1\text{.}\) [Your answer(s) will look like “\(t= \)”]
6.
Suppose the function \(p=f(t)\) represents the population of a certain town, where \(p\) represents the population and \(t\) represents the year.
Choose the equation that means, “There were \(2015\) people in the town in \(2010\text{.}\)”
\(f(2010)=2015\)
\(f(2015)=2010\)
7.
Let \(I=f(T)\) and \(H=g(T)\) be two functions that return \(I\text{,}\) the number of ice cream bars you sell at your food cart in one day, and \(H\text{,}\)the number of hot coffee drinks you sell at your food cart in one day. Let \(T\) be the average temperature outside in degrees Fahrenheit.
Do you expect the function \(f\) should be increasing or decreasing? Explain your answer.
Do you expect the function \(g\) should be increasing or decreasing? Explain your answer.
If you charge \(\$2.50\) for an ice cream bar and \(\$1.75\) for a hot coffee drink, write an expression (using function notation) that represents the total amount of money you will make from coffee and ice cream in one day, if the average temperature outside is \(80\) degrees.
8.
Let \(f(t)=78.4t9.8t^2\)
Sketch a graph of the function. Label the horizontal and vertical intercepts
Identify the intervals on which the function is increasing or decreasing.
Identify the intervals on which \(f(t)>0\text{.}\)
Identify the intervals on which the function is concave down.
Calculate \(\frac{\Delta f}{\Delta t}\)on the interval \(0 \leq t \leq 2\text{.}\)

Suppose \(f(t)\) describes the height from the ground of a model rocket in meters and \(t\) is time measured in seconds.
Based on the sketch of your graph and the situation described, what is a reasonable interval of time for this function to make sense?
The entire expression “\(\frac{\Delta f}{\Delta t}=29.4\) on the interval \(5 \leq t \leq 6\)” actually means something in English. Interpret the meaning of the expression in terms of the situation. What is the importance of the inequality as part of the expression?
9.
Arizona Cardinals running back LaRod StephensHowling once returned a kickoff to make a \(102\)yard touchdown against the Oakland Raiders. The ball was kicked from the \(30\)yard line, and it was caught \(2\) yards into the end zone. Watching the YouTube video of the kickoff, I approximated the angle at which the ball was kicked and, with the help of the internet and an old physics book, I determined the path of the ball could be modeled using the formula
where \(x\) is the horizontal distance of the ball (measured in feet) away from where the ball was kicked, and \(f(x)\) is its height above the ground (measured in feet).
Make a sketch of the path of the ball. Be sure to show the starting and ending locations of the ball, as well as the highest point. Scale each axis appropriately.
Use your graph to estimate the xlocations where the ball was \(10\) feet above the ground. Show these points on your graph. [Round your answers to the nearest foot.]

Determine the \(x\)locations where the ball was 10 feet above the ground by solving the equation
\begin{equation*} f(x)=\frac{1}{256}x^2+\frac{216}{256}x \end{equation*}algebraically. Show that your solutions are the same as those you found graphically. [Round your answers to the nearest tenth of a foot.]
10.
Suppose the function \(f(w)\) represents the hours of labor it will take for \(w\) workers to paint a particular house.
What is the best interpretation of the statement \(f(3)=120?\)
Interpret the meaning of the statement “evaluate \(f(4)\)”.
Interpret the meaning of the statement “solve \(f(w)=100\)”.
Should \(f(w)\) be an increasing or decreasing function or both? Explain how you made your conclusion.

Interpret the meaning of the statement
\begin{equation*} \frac{\Delta f}{\Delta w}=40 \, \text{on the interval} \, 1 \leq w \leq 6 \end{equation*}
11.
Let \(g(x)\) and \(f(x)\) be two functions given as formulas.
What does is mean to “Solve the equation \(g(x)=f(x)\)”?
Suppose someone claims the equation \(g(x)=f(x)\) has a solution \(x=a\text{.}\) How could you verify that \(x=a\) is actually a solution?
Graph each equation \(g(x)\) and \(f(x)\) on the same set of axes. How does one find the solution to the equation \(g(x)=f(x)\) graphically?
Let \(f(x)=x^24x\) and \(g(x)=x\text{.}\) Solve the equation \(g(x)=f(x)\) algebraically.
Let \(f(x)=x^24x\) and \(g(x)=x\text{.}\) Solve the equation \(g(x)=f(x)\) graphically. Sketch your graphs. Show and label your solutions.
12.
Let \(g(x)\) be a linear function such that \(\frac{g(5)g(2)}{3}=4\text{.}\)
Is it necessarily true that \(g(1)=4\text{?}\) Explain your answer.
Suppose we know \(g(0)=2\text{.}\) What is the value of \(g(3)\text{?}\)
Sketch an example of another function \(f(x)\) that has the same average rate of change on the interval \(2 \leq x \leq 5\) such that \(f(1) \neq 4\)
Subsection 1.4.1.2 Maintain and strengthen prerequisites, especially: percent, linear and quadratic functions, solving equations.
13.
The point \(P=(x,y)\) is on the curve \(f(x)=x^2\text{.}\)
Determine a function \(g(x)\) that calculates the distance between point \(P\) and the fixed point \((5,1)\text{.}\)
Use your function from part (a) to find the point on \(f(x)=x^2\) that is closest to the point \((5,1)\text{.}\) Use a calculator.
14.
A cylindrical tank is full of water, ready to fill a shallow swimming pool. The base of the tank has a radius of \(8\) feet, and the height is \(12\) feet. The swimming pool is a rectangular prism \(20\) feet long, \(18\) feet wide, and \(3\) feet deep.
What percent of the water in the tank will be needed to fill the pool to a depth of \(2.5\) feet?
15.
The table shows the height of the water in a graduated cylinder as a function of the volume of water in the cylinder.
Volume \((cm^3)\)  Height \((cm)\) 
\(50\)  \(0.6366\) 
\(100\)  \(1.2732\) 
\(150\)  \(1.9099\) 
\(200\)  \(2.5465\) 
\(250\)  \(3.1831\) 
Confirm that the Height is a linear function of the Volume.
Find the formula for the height as a function of the volume. Use H(V) to represent the height as a function of volume. Also, sketch a graph of the function and label two points on the graph.
Explain the meaning of the statement “\(H(1100)=14\)” in a complete sentence, including proper units.
What is the radius of the cylinder? Show how you know this.
Suppose the container is \(20 cm\) tall. What is the domain of the function \(H(V)\text{?}\)
16.
Find the area of the shaded triangular region.
17.
The figure shows a rectangle whose sides are formed by four lines. Remember that sides of a rectangle are perpendicular.
Find the area of the rectangle.
[Hint: It may help to find the equation of each side of the rectangle.]
18.
A point \(P=(x,y)\) is on the curve \(f(x)=x^2+x\text{.}\) Write the formulas for two different functions \(D(x)\) and \(S(x)\text{,}\) where the only input for each function is the xcoordinate of the point \(P\text{.}\)
\(D(x)\) finds the distance between the origin \((0,0)\) and the point P.
\(S(x)\) finds the slope of the line through the origin \((0,0)\) and the point \(P\text{.}\)
Subsection 1.4.1.3 Determine the domain and range of functions
19.
Give examples of functions with the following domains:
The set of real numbers such that \(x \geq 0\)
The set of real numbers such that \(x \neq 2\)
The set of real numbers such that \(x \neq 2\) and \(x \geq 5\)
Subsection 1.4.1.4 Exponential Functions and Constant Percent Change
20.
An investment earns \(2 \%\) each week
What is the percent change on the investment value after \(5\) weeks?
What is the overall percent change on the investment value after \(10\) weeks?
21.
A heat source is placed on a flat plate initially at \(25\) degrees Celsius. As the heat spreads across the plate the temperature of the plate increases over time. Let \(H(t)\) be the temperature at a particular point on the plate in degrees Celsius as a function of time \(t\) (measured in minutes).
Suppose \(H(5)=27\text{.}\)
Write an linear formula for \(H(t)\text{.}\)
What is \(H(50)\) if we assume \(H(t)\) is a linear function?
At what rate does the linear model predict the temperature is increasing each minute?
Write an exponential formula for \(H(t)\text{.}\)
What is \(H(50)\) if we assume \(H(t)\) is an exponential function?
At what rate does the exponential model predict the temperature is increasing each minute?
Subsection 1.4.1.5 Solve equations algebraically using properties of exponents and logarithms
Subsection 1.4.1.6 Horizontal and Vertical Translations
Subsection 1.4.1.7 Composition and Inverse
22.
A manufacturer purchases raw materials from a supplier and then makes products to sell. The amount of money the manufacturer makes from selling one of these products depends on how much they have spend on the materials. At the same time, however, the price of materials changes throughout the year.
Exstimate \(f(35)\) and interpret its meaning.
At \(40\) days after January \(1^{/rm{st}}\text{,}\) how much will they sell the product for?
Evaluate \(f(P(120))\) and interpret its meaning.
What is wrong with the composition \(P(f(20))\text{?}\) [Hint: Think about the units involved in the expression.]
Use a composition to write an equation that means: “If the company buys their materials \(280\) days after January \(1^{\rm{st}}\text{,}\) they will sell the product for \(\$ 1.19\text{.}\)”
Explain the meaning of \(f(P(0))=2\text{.}\)
Solve the equation \(f(P(t))=1\text{.}\) [Your answer(s) will look like “\(t= \)”]
23.
As the temperature drops, so does the speed of sound. The function
models the speed of sound (in miles per hour) when the temperature is \(T\) degrees Celsius. On a particular day between \(6:00\) p.m. and midnight, suppose the temperature outside is modeled by the function
where \(h\) is the number of hours after \(6{:}00\) p.m.
Determine the speed of sound at \(9\) p.m..
Determine a formula for the function that will find the speed of sound, \(h\) hours after \(6{:}00\) p.m..
24.
Let \(f(x)\) represent the number of people who visit your restaurant chain in a month, where \(x\) is the amount of money \(x\) (in dollars) that you spend on advertising. You know from experience that about \(25 \%\) of the people who visit your restaurant chain will order the daily special meal, which costs \(\$ 12.00\text{.}\)
Use function notation to write an expression that represents how much money your restaurant chain will earn in one month from the meal special if you spend \(\$ 8,000\) on advertising.
25.
On your popular YouTube channel, you earn a small amount of money (due to advertising) for each time your video is viewed.
Suppose the function \(f(x)\) represents how many times your video is viewed on the \(x^{\rm{th}}\) day of the month, and \(g(x)\) represents how much you earn per view on the \(x^{\rm{th}}\) day of the month.
Finally, suppose the function \(h(x)\) represents how much you must pay in taxes for earning \(x\) dollars from your YouTube channel.
Write an expression to represent how much money you earn on the \(5^{\rm{th}}\) day of the month.
Write an expression to represent how much you must pay in your taxes from your earnings on the \(5^{\rm{th}}\) day of the month.
26.
On a cold day, when the wind blows, it feels colder than it actually is outside. The temperature you feel is called the wind chill temperature.
The table shows values of the function \(W(s)\text{,}\) which calculates the wind chill temperature when it is \(40\) degrees Fahrenheit outside, and the wind is blowing at \(s\) miles per hour.
s, Wind Speed \((mph)\)  \(W(s)\) (degrees Fahrenheit) 
\(0\)  \(40\) 
\(10\)  \(34\) 
\(20\)  \(30\) 
\(30\)  \(28\) 
\(40\)  \(27\) 
\(50\)  \(26\) 
\(60\)  \(25\) 
Which pair of statements is true?
\(W(25)=60\) and \(W^{1}(50)=26\)
\(W(40)=27\) and \(W^{1}(34)=10\)
Explain in words the meaning of the expression \(W^{1}(30)=20\text{.}\)
Subsection 1.4.1.8 Use the characteristics of basic functions (linear, constant, polynomial, exponential, logarithmic, power, piecewise), especially slope, intercepts, rate of change, percent change and average change, to answer questions in application situations, to write equations and to create graphs by hand and on the calculator.
27.
A doctor disconnects the intravenous flow of a drug into a patient’s body at \(10\) a.m. At noon she measures the amount of drug in the patient’s body to be \(180\) mg. At \(5\) p.m. she measures \(35\) mg of drug in the patient’s body.
Find a linear formula, \(L(t)\text{,}\) that models the amount of drug, in mg, in the patient’s body as a function of time, \(t\) hours after \(10\) a.m.
Find an exponential formula, \(E(t)\text{,}\) that models the amount of drug, mg, in the patient’s body as a function of time, \(t\) hours after \(10\) a.m.
Sketch a graph of each equation \(L(t)\) and \(E(t)\) on the same set of axes. Limit the domain between \(0\) and \(12\) hours after \(10\) a.m.. Use a ruler to draw the axes and label your axes including units. (One picture, two graphs in the same picture)
Evaluate \(L(0)\) and \(E(0)\text{.}\) What do these values represent in terms of the actual situation? Explain why they must be the same or why they must be different values.
Evaluate \(L(10)\) and \(E(10)\text{.}\) What is the meaning of each result? Based on the values for each, explain which of the two formulas best models the elimination of the drug from the patient’s body?
Use your exponential equation to calculate the halflife of the drug in the body.
Calculate the hourly percent decay of the amount of drug in the body. Show your work.
Use your exponential equation to calculate \(\frac{\Delta E}{\Delta t}\) on the interval \(1 \leq x \leq 3\text{.}\) Show your work on each
28.
The function \(g(x)=kx^{2}+ \frac{1}{x}\) passes through the point \((2,5)\text{.}\) Find the value of \(k\text{.}\)
29.
The function \(p(x)=k(x+2)(x3)^{2}\) passes through the point \((2,10)\text{.}\) Find the value of \(k\text{.}\)
30.
The function \(T(x)=mx^{2}+b\) passes through the points \((0,3)\) and \((2,12)\text{.}\) Find the value of \(m\) and \(b\text{.}\)
31.
If you pull on a spring to extend it, Hooke’s Law says that the Force you must exert is directly proportional to the distance you are trying to stretch the spring.
Assign variable names to the two quantities, and write an equation involving a proportionality constant, k.
32.
Each line in the figures has a slope of \(\frac{1}{2}\text{.}\) For each graph, determine the value of \(a\) within the given coordinates.
Subsection 1.4.1.9 Recognize an exponential relationship given numerically or verbally, determine the growth/decay rate and use this information to write an equation to model the relationship
33.
At time \(t = 0\text{,}\) a right triangle initially has legs of \(3 cm\) and \(4 cm\) and a hypotenuse of \(5 cm\text{.}\) Each minute, the \(3 cm\) leg grows by \(50 \%\text{,}\) and the \(4 cm\) leg shrinks by \(50 \%\text{.}\) The shape remains a right triangle, so the hypotenuse must grow or shrink to accommodate.
Find the perimeter of the triangle after \(2\) minutes.
Find the area of the triangle after \(2\) minutes.
Use algebra to determine when the area will be exactly \(1 cm^{2}\)
34.
A circle grows so that its perimeter increases by \(50 \%\text{.}\) By what percent does its area increase?
35.
A square begins with an area of \(400 cm^{2}\text{.}\) The square grows so that its sides increase in length by \(50 \%\) per minute. What will be the area of the square after \(3\)minutes?
36.
The Rule of \(70\) says that if money (or anything else, really) is growing exponentially at a rate of \(r \%\) per year, then the number of years until the original amount will double in size is approximately \(\frac{70}{t}\text{.}\)
Choose a starting amount of money, and suppose it grows exponentially by either \(5 \%\)per year or \(12 \%\) per year. Use algebra to find how many years until the original amount of money will double in size. Then compare your answer to what the Rule of \(70\) says.
37.
In the early days of Oregon history, the town of Oregon City competed with Portland for industry and population. If things had gone differently, Portland might have stayed relatively small and Oregon City might have become the largest city in Oregon.
Now, imagine two other neighboring cities that were established in the year 1825. City \(\# 1\) begins small and grows exponentially. The town started with a group of \(50\) people, and after \(17\) years, the population had grown to \(185\) people. City \(\# 2\) starts larger than City \(\# 1\) and grows linearly. After \(6\) years, the population had grown to \(3200\text{,}\) and after \(19\) years, the population was \(5800\text{.}\)
Write equations for the population of each city. Let your independent variable \(t\) represent the number of years after \(1825\text{.}\)
What were the starting populations for the towns? What would the populations be this year?
Determine in what year the population of Town \(\# 1\) exceeds the population of Town \(\# 2\text{.}\) Explain how you determined this.