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Math Topics

Precalculus

Functions and Graphs

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1
The Coordinate System; Lines and Their Graphs

A 2012 Honda Civics value over time can be approximated by the equation \(v=-2200t+22000\) where \(t\) denotes the number of years after its purchase. Answer the following questions:

What will be the value of the car \(6\) years after purchase.

What are the slope and \(v\)-intercept, and what do they represent?

For what value of \(t\) will the value of the car be zero?

In 2010, there were 1,400 Canada Geese in a wildlife refuge. This population has been increasing by 70 geese each year.

a. Write an equation for the population of Canada Geese,p,in terms of t, the number of years since 2010.

b. What are the slope and p -intercept and what do they represent?

c. When will the goose population reach 1,960?

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2
Coordinate Geometry, Circles, and Other Equations

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3
Functions

Which of the following correspondences satisfy the definition of a function?

The input value is one of the letters from the set \(\{\text{J, F, M, A, S, O, N, D}\}\) and the output value is a name of a month beginning with that letter.

The correspondence defined by Table 1.

The input value is the radius of a circle and the output value is the area of the circle.

Eduardo is a part-time salesperson at Digitex Audio, a sound equipment store. Each week, he is paid a salary of \(\$200\) plus a commission of \(10\%\) of the amount of sales he generates that week, in dollars.

What are the input and output variables for this problem?

Express EduardoÃ¢â‚¬â„¢s pay for one week as a function of the sales he generates that week.

What was his pay for a week in which he generated \(\$4000\) worth of sales?

Table 3 gives the U.S. Postal ServiceÃ¢â‚¬â„¢s rate for a first-class letter in 2013 as a function of the weight of the letter. Letters heavier than \(3.5\) ounces are classified separately, and are not listed here.

Identify the independent variable and the dependent variable.

Explain why this table represents a function.

What is the rate for a first-class letter weighing \(3.2\) ounces?

Figure 4 depicts the average high temperature in Fargo, North Dakota, in degrees Fahrenheit, as a function of the month of the year.

Let \(T(m)\) be the function by the graph, where \(m\) is the month of the year. What is \(T(\text{May})\)?

What are the valid input values for this function?

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4
Graphs of Functions

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5
Analyzing the Graph of a Function

A ball is dropped from a height of \(100\) meters. At time \(t\), in seconds, the height of the ball from the ground is given by \(f(t)=-4.9t^2+100\). Calculate the following:

The average rate of change of \(f\) on the interval \([1,3]\).

The difference quotient \(\dfrac{f(a+h)-f(a)}{h}\)

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6
The Algebra of Functions

The GlobalEx Corporation has revenues modeled by the function \(R(t)=40+2t\), where \(t\) is the number of years since 2010 and \(R(t)\) is in millions of dollars. Its operating costs are modeled by the function \(C(t)=35+1.6t\), where \(t\) is the number of years since 2010 and \(C(t)\) is in millions of dollars. Find the profit function \(P(t)\) for GlobalEx Corporation.

Let \(f(s)=s^2+1\) and \(g(s)=-2s\).

Find an expression for \((f\circ g)(s)\) and give the domain of \(f\circ g\).

Find an expression for \((g\circ f)(s)\) and give the domain of \(g\circ f\).

Evaluate \((f\circ g)(-1)\).

Evaluate \((g\circ f)(-1)\)

The cost of fuel for running a fleet of vehicles owned by GlobalEx Corporation is given in Table 1 in terms of the number of gallons used. However, the European brand of GlobalEx Corporation records its fuel consumption in units of liters. So Table 2 lists the equivalent quantity of fuel in gallons.

a. Find the cost of 55 gallons of fuel.

b. Find the cost of the 113.55 liters of fuel.

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7
Transformations of the Graph of a Function

Make a table of values of the functions \(f(x)=\lvert x \rvert\) and \(g(x)=\lvert x \rvert -2\), for \(x=-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\). Use your table to sketch the graphs of the two functions. What are the domain and range of \(f\) and \(g\)?

Make a table of values of the functions \(f(x)=\lvert x \rvert\) and \(g(x)=\lvert x-2 \rvert\), for \(x=-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\). Use your table to sketch the graphs of the two functions. What are the domain and range of \(f\) and \(g\)?

Suppose the graph of a function \(g(x)\) is produced from the graph of \(f(x)=x^2\) by vertically compressing the graph of \(f\) by a factor of \(\dfrac{1}{3}\), then shifting it to the left by \(1\) unit, and finally shifting it downward by \(2\) units. Give an expression for \(g(x)\) and sketch the graphs of both \(f\) and \(g\).

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8
Linear Functions and Models; Variation

At the end of the year, JocelynÃ¢â‚¬â„¢s employer gives her an annual bonus of \(\$1000\) plus \(\$200\) for each year she has been employed by the company. Answer the following questions.

Find a linear function that relates JocelynÃ¢â‚¬â„¢s bonus to the number of years of her employment with the company.

Use the function you found in part (a) to calculate the annual bonus Jocelyn will receive after she has worked for the company for \(8\) years,

Use the function you found in part (a) to calculate how long Jocelyn would have to work at the company for her annual bonus to amount to \(\$3200\).

Interpret the slope and \(y\)-intercept for this problem both verbally and graphically.

Table 3 gives the value of a commercial printer at two different times after its purchase. Answer the following questions.

Identify the input and output values.

Express the value of the printer as a linear function of the number of years after its purchase.

Using the function found in part (b), find the original purchase price.

Assuming that the value of the printer is a linear function of the number of years after its purchase, when will the printerÃ¢â‚¬â„¢s value reach \(\$0\)?

Table 6 gives the body weights of laboratory rats and the weights of their hearts in grams. All data points are given to five significant digits.

Let \(x\) denote the body weight. Make a scatter plot of the heart weight \(h\) vs. \(x\).

State, in words, any general observations you can make about the data.

Find an expression for the

*linear*function that best fits the given data points.Compare the actual heart weights with the heart weights predicted by your function. What do you observe?

If a rat weighs \(308\) grams, use your model to predict its heart weight,

The volume of a fixed mass of gas is directly proportional to its temperature. If the volume of a gas is 40 cc (cubic centimeters) at 25Ã‚Â° C (degrees Celsius), find the following.

a. The variation constant k in the equation V=kT

b. The volume of the same gas at 50Ã‚Â°C.

c. The temperature of the same gas if the volume is 30 cc.

The price of a product is inversely proportional to its demand. That is, \(P=\dfrac{k}{p}\), where \(P\) is the price per unit and \(q\) is the number of products demanded. If \(3000\) units are demanded at \(\$10\) per unit, how many units are demanded at \(\$6\) per unit?