Functions

• A real-valued function f defined on a set D of real numbers is a rule that
assigns to

each number x in D exactly one real number, denoted f(x).

• The set D of all numbers for which f(x) is defined is called the domain.

• The number f(x) is called the value of the function f at the point x.

• The set of all values y = f(x) is called the range of f.

• When a function f is described by writing a formula y = f(x), we call x the
inde-

pendent variable and y the dependent variables because the value of y depends

on the choice of x.

Idea. A function is discrete if it only takes on certain isolated values. A
function is contin-

uous if it can take on any numbers.

**Domains and Intervals**

• The domain of the function f is the set of all real numbers x for which the
expression

f(x) makes sense and produces a real number y.

Often, a domain is R, or all real numbers. There are two restrictions that we
know. you

can not take the square root of a negative number and you can not divide by
zero.

• A closed interval contains both its endpoints x = a and x = b and is written
[a, b].

• A open interval contains neither of its endpoints, written (a, b).

• A half open interval contains exactly one of its endpoints, like (a, b] or [a,
b).

• An unbounded interval has positive or negative infinity as an `endpoint',
written

• The slope-intercept equation is y = mx + b for the straight line with slope
m =

angle of inclination
and y-intercept b.

• We also define the slope m as "rise over run."

• The point-slope equation for a line is y - y_{0} = m(x - x_{0}), with slope m
passing

through the point (x_{0}, y_{0}).

Horizontal lines have slope zero.

Vertical lines have no defined slope.

Parallel lines have the same slope.

**Graphs of More General Equations**

• The graph of an equation in two variables x and y is the set of all points
(x, y) in the

plane that satisfy the equation.

• The Pythagorean theorem implies the distance formula

• x^{2} + y^{2} = r^{2} is the equation for a circle of radius r centered at (0,
0).

• More generally, (x-h)^{2}+(y-k)^{2} = r^{2} is the equation for a circle of radius r
centered

at (h, k).

**Translates of Graphs**

Translation Principle. When the graph of an equation is translated h units to
the right

and k units up, the equation of the translated curve is obtained from the
original equation

by replacing x with x - h and y with y - k.

Example of completing the square to find the center of a circle, page 14.

**Graphs of Functions**

• The graph of a function is a special case of the graph of an equation.

• The graph of the function f is the graph of the equation y = f(x), so the set
of all

points in the plane (x, f(x)).

The Vertical Line Test. Each vertical line through a point in the domain of a
function

meets its graph in exactly one point.

• The values of x where the value of f(x) makes a jump are called points of
discon-

tinuity.

**Parabolas**

• The graph of a quadratic function of the form f(x) = ax^{2}
+ bx + c for a ≠ 0,
is a

parabola.

• To draw, make a table of a few points and sketch.

• The size of the coefficient a determines the width of the parabola.

• The sign on a determines if the parabola opens up or down.

• Any quadratic equation can be manipulated to be of the form y - k = a(x - h)^{2}
by

completing the square. It is now easy to see the vertex of the parabola is at
(h, k).

**Applications of Quadratic Functions**

If we plot the equations, we can find the vertex of the parabola, which
corresponds to a

maximum or minimum.

**Graphic, Numeric and Symbolic Viewpoints**

Idea. We can look at the same information graphically (in the picture of a graph
or the

sketch of a curve), numerically (like in a table of data measurements
collected), or symboli-

cally (like an equation, which gives a rule for finding any data points and
plotting them on

a graph).