Chapter 1 Preliminaries

Author

Jiaye Xu

Published

September 3, 2025

A Review

1.5 Functions and Their Graphs

1.6 Operations on Functions

1.7 Exponential and Logarithmic Functions

1.8 The Trigonometric Functions

1.9 The Inverse Trigonometric Functions

1.5 Functions and Their Graphs

Definition: A function \(f\) is a rule of correspondence that associates with each object \(x\) in one set, called the domain, a single value \(f(x)\) from a second set.The set of all values so obtained is called the range of the function.

The definition puts no restriction on the domain and range sets. The domain might consist of the set of people in your calculus class, the range the set of grades that will be given, and the rule of correspondence the assignment of grades. Nearly all functions you encounter in this book will be functions of one or more real numbers.

N.B. A function \(f\) is a rule that assigns a unique value \(f(x)\) to each \(x\).

Domain and Range

To specify a function completely, we must state, in addition to the rule of correspondence, the domain of the function.

When no domain is specified for a function, we assume that it is the largest set of real numbers for which the rule for the function makes sense. This is called the natural domain.

Graphs of Functions

When both the domain and range of a function are sets of real numbers, we can picture the function by drawing its graph on a coordinate plane.

Sketch the graph by hand and via computing tools. For example, \(y=f(x)=x^2-2\)

x <- seq(-3, 3, by=0.1)
y <- x^2-2
plot(x,y, type = "l")

# equivalently,
curve(x^2-2, from = -3, to = 3)

To sketch the graph of \(g(x)=2/(x-1)\),

curve(2/(x-2), from = -2, to = 4, lwd=2)
abline(h=0,v=0,  lty="dashed", col="blue")

The Vertical Line Test for a Function

Not every curve in the coordinate plane can be the graph of a function.

N.B. A function \(f\) can have only one value \(f(x)\) for each \(x\) in its domain, so no vertical line can intersect the graph of a function more than once.

If \(a\) is in the domain of the function \(f\), then the vertical line \(x = a\) will intersect the graph of \(f\) at the single point \((a, f(a))\).

For example, a circle cannot be the graph of a function, since some vertical lines intersect the circle twice.

The unit circle graphed in the figure below, however, contains the graphs of two functions of \(x\), namely the upper semicircle defined by the function \(f(x) = \sqrt{1 - x^2}\) and the lower semicircle defined by the function \(g (x) = -\sqrt{1 - x^2}\)

Even and Odd Functions: Symmetry of graph

We can often predict the symmetries of the graph of a function by inspecting the formula for the function.

If \(f(-x)=f(x)\) for all \(x\), then the graph is symmetric with respect to the \(y\)-axis. Such a function is called an even function.
If \(f(-x)=-f(x)\) for all \(x\), the graph is symmetric with respect to the origin. We call such a function an odd function.

Increasing, Decreasing, Monotonic Functions

Definition: Let \(I\) be an interval containing the points \(x_1\) and \(x_2\). A function \(f\) is increasing on \(I\) if \(x_1 < x_2\) implies \(f(x_1) < f(x_2)\). A function \(f\) is decreasing on \(I\) if \(x_1 < x_2\) implies \(f(x_1) > f(x_2)\). A function \(f\) is monotonic on \(I\) if it is either increasing or decreasing on \(I\).

Two Special Functions

The Absolute Value Functions
The Greatest Integer Functions

They are defined by

1.6 Operations on Functions

Sums, Differences, Products, Quotients, and Powers

We must be a little careful about domains. For example, the domain of \(f+g\) is the intersection (common part) of the domains of \(f\) and \(g\). This intersection rule for the domain also holds for other operations.

Consider functions \(f\) and \(g\) with formulas \[f(x)=\frac{x-3}{2}, \quad g(x)=\sqrt{x},\] assuming they have their natural domains, we have the following:

N.B. We had to exclude \(0\) from the domain of \(f/g\) to avoid division by \(0\).

Composition of Functions

If \(f\) works on \(x\) to produce \(f(x)\) and \(g\) then works on \(f(x)\) to produce \(g(f(x))\), we say that we have composed \(g\) with \(f\).The resulting function, called the composition of \(g\) with \(f\), is denoted by \(g\circ f\) Thus, \[(g\circ f) (x)=g(f(x))\]

We must be careful in describing the domain of a composite function. The domain of \(g\circ f\) is equal to the set of those values \(x\) that satisfy the following properties (figure 3, page 36):

\(x\) is in the domain of \(f\).
\(f(x)\) is in the domain of \(g\).

N.B. The domains of \(g\circ f\) and \(f\circ g\) can be different.

The decomposition of a given function. It is your turn to write a function as a composition of function \(g\) with \(f\), for example, \(p(x)=(x+2)^5\).

The Inverse of a Function

In some cases, we can reverse \(f\); that is, for any given \(y\) in the range \(R\), we can unambiguously go back and find the \(x\) from which it came. This new function that takes \(y\) and assigns \(x\) to it is denoted by \(f^{-1}\). The function \(f^{-1}\) is called the inverse of \(f\) or simply \(f\)-inverse.

Note that its domain is \(R\) (the range of \(f\) and its range is \(D\) (the domain of \(f\)).

N.B. Symbol \(f^{-1}\) means inverse not reciprocal.

Geometrically, a function that satisfies the horizontal line test will have an inverse. That is, a horizontal line can intersect the graph of \(y=f(x)\) at most once. Equivalently, we define the one-to-one function,

Definition: A function \(f\) is said to be one-to-one if distinct values of \(x\) always lead to distinct values of \(f(x)\); that is \[x_1\neq x_2 \iff f(x_1) \neq f(x_2)\]
Theorem A: If \(f\) is monotonic on its domain, then \(f\) has an inverse.

The Graph of \(y=f^{-1}(x)\)

The graph of \(y=f^{-1}(x)\) is just the reflection of the graph of \(y=f(x)\) across the line \(y=x\) (Figure 14 & 15, page 40).

Partial Catalog of Functions

Constant function \(f(x)=k\)
Identity function \(f(x)=x\)
- explicit algebraic function: obtained from the constant functions and the identity function via the five operations of addition, subtraction, multiplication, division, and root extraction. e. g., \(y=kx^2\)
Polynomial function (n-th degree) \[f(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0\]
- linear function (1st degree)
- quadratic function (2nd degree)
Transcendental functions. These include the trigonometric, inverse trigonometric, exponential, and logarithmic functions.

1.7 Exponential and Logarithmic Functions

Graphs of Exponential Functions

Property of Exponents

Logarithmic Functions

Base Changing Formula for the logarithms,

The common logarithm is the logarithm to the base 10. Another base that will become important in the next chapter is the natural logarithm, that is the logarithm to the base \(e\approx 2.718\).

1.8 The Trigonometric Functions

More generally, we define the trigonometric functions based on the unit circle (Figure 2, page 51). There is a single point \(P\) on the circle \(C\) such that the distance, measured in the counterclockwise direction around the arc \(AP\), is equal to \(t\).

Period and Amplitude of the Trigonometric Functions

A function \(f\) is periodic if there is a positive number \(p\) such that \[f(x+p)=f(x)\] for all real numbers \(x\) in the domain of \(f\). If \(f\) is nonconstant, the smallest such positive number \(p\) is called the period of \(f\).

For example, if let \(f(x)=\sin (ax)\), since we know \(\sin (ax)=\sin(ax+2\pi)\), and \(\sin\left(a(x+\frac{2\pi}{a})\right)=f(x+\frac{2\pi}{a})\), then \(f(x+\frac{2\pi}{a})=f(x)\). The period of the function \(\sin (ax)\) is \(\frac{2\pi}{a}\).

If the periodic function \(f\) attains a minimum and a maximum, we define the amplitude \(A\) to be half the vertical distance between the highest point and the lowest point on the graph.

Four Other Trigonometric Functions

Trigonometric Identities

1.9 The Inverse Trigonometric Functions

Restricting the Domain

We are going to introduce a notion of inverse for the trigonometric functions. That this is possible rests on a procedure called restricting the domain as shown in Figure 3 on page 61.