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1 | FUNCTIONS AND GRAPHS
Chapter Outline
Introduction
1.1 | Review of Functions
Learning Objectives
Functions
In this case, the domain is considered to be the set of all real numbers x for which f(x) is a real number. To denote the set of non-negative real numbers, we would use the set-builder notation.
Evaluating Functions
When this function is evaluated for an input x, the equation to use depends on whether x ≥ 2 or x < 2.
Finding Domain and Range
To show that every element in this set is in the range of f , we need to show that for all y in this set, there exists a real number x in the domain such that f (x) = y. To find the range of f , we need to find values of y such that there exists a real number x in the domain with the property that.
Representing Functions
On the other hand, if c is not in the domain of f , f (c) is not defined and the line x = c does not cut the graph of f. We can use this test to determine whether a set of graphed points represents the graph of a function (Figure 1.8).
Finding Zeros and y -Intercepts of a Function
Using Zeros and y -Intercepts to Sketch a Graph
Using the table and knowing that, since the function is a square root, the graph of f must be similar to the graph of y = x, we sketch the graph (Figure 1.10).
Finding the Height of a Free-Falling Object
Note that for this function and the function f(x) = −4x + 2 shown in Figure 1.9, the values of f (x) become smaller as x becomes larger. For example, using our temperature function in Figure 1.6, we can see that the function is decreasing on the interval (0, 4), increasing on the interval (4, 14), and then decreasing on the interval (14, 23).
Combining Functions
To combine functions using mathematical operators, we simply write the functions with the operator and simplify them.
Combining Functions Using Mathematical Operations
We note that since cost is a function of temperature and temperature is a function of time, it makes sense to define this new function (C∘T)(t). First, the function f maps each input x in the domain of f to its output f(x) in the range of f.
Compositions of Functions Defined by Formulas
Second, since the range of f is a subset of the domain of g, the output f (x) is an element in the domain of g, and is therefore mapped to an output g⎛⎝f(x)⎞⎠ on the range of g. If y is in the interval (0, 1], the expression under the radical is nonnegative, and therefore there exists a real number x such that 1/(x2+ 1) = y.
Composition of Functions Defined by Tables
Application Involving a Composite Function
Symmetry of Functions
If a function f has this property, we say that f is an even function, which has symmetry about the y-axis. If f has this property, we say that f is an odd function, which has symmetry about the origin.
Even and Odd Functions
Looking again at Figure 1.14, we see that since f is symmetric about the y-axis, if the point (x, y) is on the graph, the point (−x, y) is on the graph. Conversely, looking again at Figure 1.14, if a function f is symmetric about the origin, then whenever the point (x, y) is on the graph, the point (−x, −y) is also on the graph.
Working with the Absolute Value Function
EXERCISES
For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. Find the exact and two-digit approximation of the length of the sides of a square with an area of 56 square units.
1.2 | Basic Classes of Functions
Linear Functions and Slope
Since we often use the symbol m to denote the slope of a line, we can write f (x) = mx + b. Suppose that the graph of a linear function passes through the point (x1, y1) and the slope of the line is m.
Finding the Slope and Equations of Lines
To find an equation for a linear function in point-slope form, use the slope m = −3/5 and pick any point on the line. To find an equation for a linear function in slope form, solve the equation in part b.
A Linear Distance Function
To draw D, use the fact that the graph passes through the origin and has a slope m = 3/26.
Polynomials
The line y = 2 is a horizontal asymptote for the function f(x) = 2 + 1/x because the graph of the function gets closer to the line as x increases. Suppose the data in Table 1.6 shows the number of units a company sells as a function of price per item.
Maximizing Revenue
Algebraic Functions
Find Domain and Range for Algebraic Functions For each of the following functions, find the domain and range.
Finding Domain and Range for Algebraic Functions For each of the following functions, find the domain and range
Finding Domains for Algebraic Functions
Transcendental Functions
Classifying Algebraic and Transcendental Functions
Piecewise-Defined Functions
Parking Fees Described by a Piecewise-Defined Function
Write a piecewise defined function that describes the cost C to park in the parking garage as a function of x hours parked. Write a piecewise defined function that describes the cost C as a function of weight x for 0 < x ≤ 3, where C is measured in cents and x is measured in ounces.
Transformations of Functions
We investigated what happens to the graph of a function f when we multiply f by a constant c > 0 to get a new function c f (x). However, we have not addressed what happens to the graph of the function if the constant c is negative.
Transforming a Function
EXERCISES
For the following exercises, write the equation of the line that satisfies the given conditions in slope form. At the end of the 3-year period, the value of the equipment was depreciated on a straight-line basis to $12,300.
1.3 | Trigonometric Functions
Radian Measure
Converting between Radians and Degrees a. Express 225° using radians
The Six Basic Trigonometric Functions
The values of the trigonometric functions for θ are defined in terms of the coordinates x and y. The values of the other trigonometric functions can be expressed in terms of x, y and r (Figure 1.32).
Evaluating Trigonometric Functions
The values of the other trigonometric functions are easily calculated from the values of sinθ and cosθ. As mentioned earlier, the ratios of the side lengths of a right triangle can be expressed in terms of the trigonometric functions evaluated at any of the acute angles of the triangle.
Constructing a Wooden Ramp
Let A be the length of the adjacent leg, O the length of the opposite leg and H the length of the hypotenuse. If the angle between the base of the ladder and the ground is to be 60°, how far from the house should she place the base of the ladder.
Trigonometric Identities
Solving Trigonometric Equations
To solve this equation, it is important to note that we must factor the left side and not divide both sides of the equation by cosθ. Returning to the equation, it is important to factor sinθ out of both terms on the left side rather than dividing both sides of the equation by sinθ.
Proving a Trigonometric Identity
A shifted sine curve occurs naturally when we graph the number of hours of daylight at a given location as a function of the day of the year. For example, the city reports that June 21 is the longest day of the year with 15.7 hours, and December 21 is the shortest day of the year with 8.3 hours.
Sketching the Graph of a Transformed Sine Curve
EXERCISES
For the following exercises, each graph is of the form y = AsinBx or y = AcosBx, where B > 0. 12(t − 8)⎤⎦ is a mathematical model of the temperature (in degrees Fahrenheit) atthours after midnight at a given time . day of the week.
1.4 | Inverse Functions
Existence of an Inverse Function
One way to determine whether a function is one-to-one is to look at its graph. A function f is one-to-one if and only if every horizontal line intersects the graph of f no more than once.
Determining Whether a Function Is One-to-One
Since the horizontal line y = n for every integer n ≥ 0 intersects the graph more than once, this function is not one-to-one. Since each horizontal line intersects the graph once (at most), this function is one-to-one.
Finding a Function’s Inverse
Finding an Inverse Function
Let us consider the relationship between the graph of a function f and the graph of its inverse.
Sketching Graphs of Inverse Functions
Using the previous strategy for finding inverse functions, we can verify that the inverse function is f−1(x) = x2− 2, as shown in the graph. Sketch the graph of f(x) = 2x + 3 and the graph of its inverse, using the symmetry property of inverse functions.
Restricting the Domain
Inverse Trigonometric Functions
To graph the inverse trigonometric functions, we use the graphs of the trigonometric functions restricted to the previously defined domains and plot the graphs around the line y = x (Figure 1.41). The problem is that the inverse sine function, sin−1, is the inverse of the restricted sine function defined on the domain ⎡⎣−π2, π.
Evaluating Expressions Involving Inverse Trigonometric Functions
The inverse function is supposed to "undo" the original function, so why isn't sin−1⎛⎝sin(π)⎞⎠= π. Recalling our definition of inverse functions, the function f and its inverse f−1 satisfy the conditions f⎛⎝f−1(y)⎞⎠= y for all y in the domain f−1 and f−1⎛⎝f( x)⎞⎠ = x for all x in the domain of f, so what happened here.
The Maximum Value of a Function
EXERCISES
Find the distance from the center of an artery with speeds of 15 cm/sec, 10 cm/sec and 5 cm/sec. T]The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function.
1.5 | Exponential and Logarithmic Functions
Exponential Functions
We claim that as we choose rational numbers x that get closer and closer to 2, the values of 2x get closer and closer to a number L. For more research on the graphs of exponential functions, visit this site (http://www.openstaxcollege.org/l/20_inverse).
Using the Laws of Exponents
If the money is compounded twice a year, the amount of money after half a year is A⎛⎝1. More generally, if the money is compounded once a year, the amount of money in the account after t years is given by the function.
Compounding Interest
Logarithmic Functions
Using this fact and the graphs of the exponential functions, we plot functions logb for various values of b > 1 (Figure 1.47). Before we solve some equations involving exponential and logarithmic functions, let's review the basic properties of logarithms.
Solving Equations Involving Exponential Functions Solve each of the following equations for x
Solving Equations Involving Logarithmic Functions
If you need to use a calculator to evaluate an expression with a different base, you can apply the change-of-base formulas first. For the first change-of-base formula, we start by using the power property of logarithmic functions.
Chapter Opener: The Richter Scale for Earthquakes
How can we use logarithmic functions to compare the relative severity of the magnitude 9 earthquake in Japan in 2011 and the magnitude 7.3 earthquake in Haiti in 2010. Therefore, A1/A2= 101.7 and we conclude that the earthquake in Japan was about 50- times larger than the earthquake in Haiti.
Hyperbolic Functions
Using the definition of cosh(x) and the principles of physics, it can be shown that the height of a suspended chain, such as the one in Figure 1.49, can be described by the function h(x) = acosh(x/a) + c for certain constants a and c. The graphs of the other three hyperbolic functions can be sketched using the graphs of coshx, sinhx, and tanhx (Figure 1.51).
Evaluating Hyperbolic Functions a. Simplify sinh(5lnx)
Evaluating Inverse Hyperbolic Functions Evaluate each of the following expressions
EXERCISES
For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal point. For the following exercises, use the change of base formula and either base 10 or basee to evaluate the given expressions.
REVIEW
KEY TERMS
KEY EQUATIONS
KEY CONCEPTS
REVIEW EXERCISES
For the following problems, determine the largest domain on which the function is one-to-one, and find the inversion on that domain. For the following problems, consider the seasonally cyclical population of Ocean City, New Jersey.
2 | LIMITS
This chapter was created in an informal, intuitive way, but this is not always enough when we need to prove a mathematical statement involving limits. The last section of this chapter presents the more precise definition of a limit and shows how to prove whether a function has a limit.
2.1 | A Preview of Calculus
Next, we describe how to find the limit of a function at a given point.
The Tangent Problem and Differential Calculus
The accuracy of approximating the rate of change of the function with a secant line depends on how close to the function. The slope of the tangent line to the graph ata measures the rate of change of the function ata.
Finding Slopes of Secant Lines
Can we use the same ideas to create a reasonable definition of the instantaneous velocity at a given time t = a. Furthermore, to find the slope of a tangent line at a point, we let the x values be closer to the slope of the secant line.
Finding Average Velocity
Similarly, to find the instantaneous velocity at timea, we let the t-values approximate the average velocity. Estimate its instantaneous velocity at time t = 2 by calculating its average velocity over the time interval.
The Area Problem and Integral Calculus
Estimation Using Rectangles
Other Aspects of Calculus
EXERCISES
Use the value in the preceding exercise to find the equation of the tangent at point P. Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent to x = 1.
2.2 | The Limit of a Function
Intuitive Definition of a Limit
If values of the function f(x) approach the real number If the values of x( ≠ a) approach the number, then we say that the limit of f(x) asx approaches L. We can estimate limits by constructing tables of functional values and by looking at their graphs.
Evaluating a Limit Using a Table of Functional Values 1
Using a graphing calculator or computer software that allows us to graph functions, we can plot the function f(x), and make sure that the function values of f(x) forx values in our window are close. We can use the trace feature to move along the graph of the function and see how their value is read as the x values approach.
Evaluating a Limit Using a Table of Functional Values 2 Evaluate x → 4lim x − 2
At this point, we see from Example 2.4 and Example 2.5 that it can be just as easy, if not easier, to evaluate a limit of a function by inspecting its graph as it is to evaluate the limit using a table of values functional.
Evaluating a Limit Using a Graph
Two Important Limits Let a be a real number and c be a constant
Note that for all values of x (whether or not they approach), the values of f(x) remain constant atc.
The Existence of a Limit
Evaluating a Limit That Fails to Exist
One-Sided Limits
Relating One-Sided and Two-Sided Limits
Infinite Limits
Recognizing an Infinite Limit
We must also emphasize that in the graphs f(x) = 1/(x − a)n the points on the graph that have x-coordinates very close to a are very close to the vertical line x = a.
Finding a Vertical Asymptote
Behavior of a Function at Different Points
Chapter Opener: Einstein’s Equation
EXERCISES
In the following exercises, use the graph of the function y = f (x) shown here to find the values, if possible. In the following exercises, use the graph of the function y = g(x) shown here to find the values, if possible.
2.3 | The Limit Laws
Evaluating Limits with the Limit Laws
Basic Limit Results For any real number a and any constant c,
Evaluating a Basic Limit
Limit Laws
Evaluating a Limit Using Limit Laws
Using Limit Laws Repeatedly
Again, we must keep in mind that as we rewrite the limit in terms of other limits, each new limit must exist for the limit law to apply.
Limits of Polynomial and Rational Functions
Additional Limit Evaluation Techniques
In this case, we say that f(x)/g(x) has the indefinite form 0/0.) The following problem-solving strategy provides a general overview for evaluating these types of limits. The following examples demonstrate the use of this problem-solving strategy. Example 2.17 illustrates the factor-and-cancel technique; Example 2.18 shows multiplying by a conjugate.
Evaluating a Limit by Simplifying a Complex Fraction
Evaluating a Limit When the Limit Laws Do Not Apply Evaluate x → 0lim ⎛
Evaluating a One-Sided Limit Using the Limit Laws Evaluate each of the following limits, if possible
Example 2.20 does not fit well into any of the patterns established in the previous examples. Since this function is not defined to the left of 3, we cannot apply limit laws to calculate x → 3−lim x − 3.
The Squeeze Theorem
The Squeeze Theorem
Applying the Squeeze Theorem
We see that the length of the side opposite angle θ in this new triangle is tanθ. Dividing by sinθ in all parts of the inequality, we get 1 < θsinθ < 1.
Evaluating an Important Trigonometric Limit Evaluate θ → 0lim 1 − cosθ
Using a manipulation similar to that used in proving that θ → 0−lim sinθ = 0, we can show that θ → 0−lim sinθ.
Deriving the Formula for the Area of a Circle
EXERCISES
In the following exercises, use direct substitution to show that each limit leads to the undefined form 0/0. In the following exercises, use the graphs below and the limit laws to evaluate each limit.
2.4 | Continuity
Continuity at a Point
Now we put our list of conditions together and form a definition of continuity at a point. If x → alim f(x) does not exist (that is, it is not a real number), then the function is not continuous and the problem is solved.
Determining Continuity at a Point, Condition 1
The next three examples show how to use this definition to determine whether a function is continuous at a given point.
Determining Continuity at a Point, Condition 2
Continuity of Polynomials and Rational Functions Polynomials and rational functions are continuous at every point in their domains
Continuity of a Rational Function
Types of Discontinuities
Classifying a Discontinuity
Continuity over an Interval
Continuity on an Interval
The proof of the next theorem uses the composite function theorem as well as the continuity of f(x) = sinx and g(x) = cosx at the point 0 to show that the trigonometric functions are continuous in all their domains.
Continuity of Trigonometric Functions Trigonometric functions are continuous over their entire domains
Proof
The Intermediate Value Theorem
The Intermediate Value Theorem
Application of the Intermediate Value Theorem
When Can You Apply the Intermediate Value Theorem?
EXERCISES
For the following exercises, determine the points, if any, at which each function is discontinuous. In the following exercises, find the value(s) of e that makes each function continuous over the given interval.
