Supplement: Review of Elementary Functions
In this supplement, we review the basic features and characteristics of the
simple elementary functions.
S.1 Affine Functions
By an affine function, we mean an expression of the form
y(x) = ax + b (S.1)
In the special case where b = 0, we say that y is a linear function of x.
We can interpret the parameters in the above function as representing the
slope-intercept form of a straight line. Here, a is the slope, which is a measure
of the steepness of a line; and b is the y-intercept (i.e., the line intersects the
y-axis at the point (0, b)).
The following cases illustrate the different possibilities:
1. a = 0: this specifies a horizontal line at a height b above the x-axis
and that has zero slope.
2. a > 0: the height of a point on the line (i.e., the y-value) increases
as the value of x increases.
3. a < 0: the height of the line decreases as the value of x increases.
4. b > 0: the line y-intercept is positive.
5. b < 0: the line y-intercept is negative.
6. x = k: this function represents a vertical line passing through the
point (k, 0).
It should be noted that:
" If two lines have the same slope, they are parallel.
" Two nonvertical lines are perpendicular if and only if their slopes
are negative reciprocals of each other. (It is easy to deduce this
0-8493-????-?/00/$0.00+$.50
© 2000 by CRC Press LLC
© 2001 by CRC Press LLC
property if you remember the relationship that you learned in
trigonometry relating the sine and cosine of two angles that differ
by Ä„/2.) See Section S.4 for more details.
FIGURE S.1
Graph of the line y = ax + b (a = 2, b = 5).
S.2 Quadratic Functions
Parabola
A quadratic parabolic function is an expression of the form:
y(x) = ax2 + bx + c where a `" 0 (S.2)
Any x for which ax2 + bx + c = 0 is called a root or a zero of the quadratic func-
tion. The graphs of quadratic functions are called parabolas.
If we plot these parabolas, we note the following characteristics:
1. For a > 0, the parabola opens up (convex curve) as shown in
Figure S.2.
2. For a < 0, the parabola opens down (concave curve) as shown in
Figure S.2.
© 2001 by CRC Press LLC
FIGURE S.2
Graph of a quadratic parabolic (second-order polynomial) function with 0 or 2 roots.
3. The parabola does not always intersect the x-axis; but where it
does, this point s abscissa is a real root of the quadratic equation.
A parabola can cross the x-axis in either 0 or 2 points, or the x-axis can be
tangent to it at one point. If the vertex of the parabola is above the x-axis and
the parabola opens up, there is no intersection, and hence, no real roots. If, on
the other hand, the parabola opens down, the curve will intersect at two val-
ues of x equidistant from the vertex position. If the vertex is below the x-axis,
we reverse the convexity conditions for the existence of two real roots. We
recall that the roots of a quadratic equation are given by:
-b Ä… b2 - 4ac
xÄ… = (S.3)
2a
When b2  4ac < 0, the parabola does not intersect the x-axis. There are no
real roots; the roots are said to be complex conjugates. When b2  4ac = 0, the
x-axis is tangent to the parabola and we have one double root.
Geometrical Description of a Parabola
The parabola can also be described through the following geometric con-
struction: a parabola is the locus of all points P in a plane that are equidistant
from a fixed line (called the directrix) and a fixed point (called the focus) not
situated on the line.
© 2001 by CRC Press LLC
FIGURE S.3
Graph of a parabola defined through geometric parameters. (Parameter values: h = 2, k =
2, p = 1.)
d1 = d2 (S.4)
The algebraic expression for the parabola, using the above geometric
parameters, can be obtained by specifically writing and equating the expres-
sions for the distances of a point on the parabola from the focus and from the
directrix:
(x - h)2 + (y - (k + p))2 = y - (k - p) (S.5)
Squaring both sides of this equation, this equality reduces to:
(x - h)2 = 4p(y - k) (S.6)
or in standard form, it can be written:
ëÅ‚ öÅ‚
x2 h h2 + 4pk
y = - x + (S.7)
ìÅ‚ ÷Å‚
4p 2p 4p
íÅ‚ Å‚Å‚
© 2001 by CRC Press LLC
Ellipse
The standard form of the equation describing an ellipse is given by:
(y
(x - h)2 - k)2
+ = 1 (S.8)
a2 b2
The ellipse s center is located at (h, k), and assuming a > b, the major axis
length is equal to 2a, the minor axis length is equal to 2b, the foci are located
at (h  c, k) and (h + c, k), and those of the vertices at (h  a, k) and (h + a, k);
where
c2 = a2  b2 (S.9)
Geometric Definition of an Ellipse
An ellipse is the locus of all points P such that the sum of the distance
between P and two distinct points (called the foci) is constant and greater
than the distance between the two foci.
d1 + d2 = 2a (S.10)
The center of the ellipse is the midpoint between foci, and the two points of
intersection of the line through the foci and the ellipse are called the vertices.
The eccentricity of an ellipse is the ratio of the distance between the center
and a focus over the distance between the center and a vertex; that is
µ = c/a (S.11)
FIGURE S.4
Graph of an ellipse defined through geometric parameters. (Parameter values: h = 2, k = 2,
a = 3, b = 2.)
© 2001 by CRC Press LLC
Hyperbola
The standard form of the equation describing a hyperbola is given by:
(y
(x - h)2 - k)2
- = 1 (S.12)
a2 b2
The center of the hyperbola is located at (h, k), and assuming a > b, the major
axis length is equal to 2a, the minor axis length is equal to 2b, the foci are
located at (h  c, k) and (h + c, k), and those of the vertices at (h  a, k) and (h +
a, k). In this case, c > a > 0 and c > b > 0 and
c2 = a2 + b2 (S.13)
Geometric Definition of a Hyperbola
A hyperbola is the locus of all points P in a plane such that the absolute value
of the difference of the distances between P and the two foci is constant and
is less than the distance between the two foci; that is
d1 - d2 = 2a (S.14)
FIGURE S.5
Graph of a hyperbola defined through geometric parameters. (Parameter values: h = 2, k =
2, a = 1, b = 3.)
© 2001 by CRC Press LLC
The center of the hyperbola is the midpoint between foci, and the two points
of intersection of the line through the foci and the hyperbola are called the
vertices.
S.3 Polynomial Functions
A polynomial function is an expression of the form:
p(x) = anxn + an-1xn-1 + & + a1x + a0 (S.15)
where an `" 0 for an nth degree polynomial.
The Fundamental Theorem of Algebra states that, for the above polyno-
mial, there are exactly n complex roots; furthermore, if all the polynomial
coefficients are real, then the complex roots always come in pairs consisting
of a complex number and its complex conjugate.
S.4 Trigonometric Functions
The trigonometric circle is defined as the circle with center at the origin of the
coordinates axes and having radius 1.
The trigonometric functions are defined as functions of the components of
a point P on the trigonometric circle. Specifically, if we define the angle ¸ as
the angle between the x-axis and the line OP, then:
" cos(¸) is is the x-component of the point P.
" sin(¸) is the y-component of the point P.
Using the Pythagorean theorem in the right angle triangle OQP, one
deduces that:
sin2(¸) + cos2(¸) = 1 (S.16)
Using the above definitions for the sine and cosine functions and elementary
geometry, it is easy to note the following properties for the trigonometric
functions:
sin(-¸) = - sin(¸) and cos(-¸) = cos(¸) (S.17)
sin(¸ + Ä„) = - sin(¸) and cos(¸ + Ä„) = - cos(¸) (S.18)
© 2001 by CRC Press LLC
FIGURE S.6
The trigonometric circle.
sin(¸ + Ä„ / 2) = cos(¸) and cos(¸ + Ä„ / 2) = - sin(¸) (S.19)
sin(Ä„ / 2 - ¸) = cos(¸) and cos(Ä„ / 2 - ¸) = sin(¸) (S.20)
The tangent and cotangent functions are defined as:
sin(¸) 1
tan(¸) = and cot(¸) =(S.21)
cos(¸) tan(¸)
Other important trigonometric relations relate the angles and sides of a tri-
angle. These are the so-called Law of Cosines and Law of Sines in a triangle:
c2 = a2 + b2 - 2ab cos(Å‚) (S.22)
sin(Ä…) sin(²) sin(Å‚)
= = (S.23)
a b c
where the sides of the triangle are a, b, c, and the angles opposite, respectively,
of each of these sides are denoted by Ä…, ², Å‚.
© 2001 by CRC Press LLC
S.5 Inverse Trigonometric Functions
 1
The inverse of a function y = f(x) is a function, denoted by x = f (y), having
 1
the property that y = f(f (y)). It is important to note that a function f(x) that
is single-valued (i.e., to each element x in its domain, there corresponds one,
and only one, element y in its range) may have an inverse that is multi-valued
(i.e., many x values may correspond to the same y). Typical examples of
multi-valued inverse functions are the inverse trigonometric functions. In
such instances, a single-valued inverse function can be defined if the range of
the inverse function is defined on a more limited region of space. For exam-
ple, the cos 1 function (called arc cosine) is single-valued if 0 d" x d" Ä„.
Note that the above notation for the inverse of a function should not be con-
fused with the negative-one power of the function f(x), which should be writ-
ten as:
(f(x)) 1 or 1/f(x)
Also note that because the inverse function reverses the role of the x- and
y-coordinates, the graphs of y = f(x) and y = f 1(x) are symmetric with respect
to the line y = x (i.e., the first bisector of the coordinate axes).
S.6 The Natural Logarithmic Function
The natural logarithmic function is defined by the following integral:
x
1
ln(x) = dt (S.24)
+"
t
1
The following properties of the logarithm can be directly deduced from the
above definition:
ln(ab) = ln(a) + ln(b) (S.25)
ln(ar ) = r ln(a) (S.26)
1
lnëÅ‚ öÅ‚ =- ln(a) (S.27)
íÅ‚
ałł
a
lnëÅ‚ öÅ‚ = ln(a) - ln(b) (S.28)
íÅ‚
błł
© 2001 by CRC Press LLC
To illustrate the technique for deriving any of the above relations, let us
consider the first of them:
ab a ab
1 1 1
ln(ab) = dt = dt + dt (S.29)
+" +" +"
t t t
11 a
The first term on the RHS is ln(a), while the second term through the substi-
tution u = t/a reduces to the definition of ln(b).
Note that:
ln(1) = 0 (S.30)
ln(e) = 1 (S.31)
where e = 2.71828.
S.7 The Exponential Function
The exponential function is defined as the inverse function of the natural log-
arithmic function; that is
exp(ln(x)) = x for all x > 0 (S.32)
ln(exp(y)) = y for all y (S.33)
The following properties of the exponential function hold for all real numbers:
exp(a)exp(b) = exp(a + b) (S.34)
(exp(a))b = exp(ab) (S.35)
1
exp(-a) = (S.36)
exp(a)
exp(a)
= exp(a - b) (S.37)
exp(b)
It should be pointed out that any of the above properties can be directly
obtained from the definition of the exponential function and the properties of
© 2001 by CRC Press LLC
the logarithmic function. For example, the first of these relations can be
derived as follows:
ln(exp(a)exp(b)) = ln(exp(a)) + ln(exp(b)) = a + b (S.38)
Taking the exponential of both sides of this equation, we obtain:
exp(ln(exp(a)exp(b))) = exp(a)exp(b) = exp(a + b) (S.39)
which is the desired result.
Useful Features of the Exponential Function
If the exponential function is written in the form:
y(x) = exp(-bx) (S.40)
the following features are apparent:
1. If b > 0, then the function is convergent at (+ infinity) and goes to
zero there.
2. If b < 0, then the function blows up at (+ infinity).
3. If b = 0, then the function is everywhere equal to a constant y = 1.
4. The exponential functions are monotonically increasing for b < 0,
and monotonically decreasing for b > 0.
5. If b1 > b2 > 0, then everywhere on the positive x-axis, y1(x) < y2(x).
6. The exponential function has no roots.
7. For b > 0, the product of the exponential function with any poly-
nomial goes to zero at (+ infinity).
We plot in Figures S.7 and S.8 examples of the exponential function for dif-
ferent values of the parameters. The first six properties above are clearly
exhibited in these figures.
S.8 The Hyperbolic Functions
The hyperbolic cosine function is defined by:
exp(x) + exp(-x)
cosh(x) = (S.41)
2
© 2001 by CRC Press LLC
FIGURE S.7
The graph of the function y = exp( bx), for different positive values of b.
FIGURE S.8
The graph of the function y = exp( bx), for different negative values of b.
© 2001 by CRC Press LLC
and the hyperbolic sine function is defined by:
exp(x) - exp(-x)
sinh(x) = (S.42)
2
Using the above definitions, it is straightforward to derive the following
relations:
cosh2(x) - sinh2(x) = 1 (S.43)
1 - tan2(x) = sech2(x) (S.44)
S.9 The Inverse Hyperbolic Functions
y = sinh-1(x) if x = sinh(y) (S.45)
Using the definition of the hyperbolic functions, we can write the inverse
hyperbolic functions in terms of logarithmic functions. For example, consid-
ering the inverse hyperbolic sine function from above, we obtain:
ey - 2x - e-y = 0 (S.46)
multiplying by ey everywhere, we obtain a second-degree equation in ey:
e2y - 2xey - 1 = 0 (S.47)
Solving this quadratic equation, and choosing the plus term in front of the
discriminant, since ey is everywhere positive, we obtain:
ey = x + x2 + 1 (S.48)
giving, for the inverse hyperbolic sine function, the expression:
y = sinh-1(x) = ln(x + x2 + 1) (S.49)
In a similar manner, one can show the following other identities:
cosh-1(x) = ln(x + x2 - 1) (S.50)
© 2001 by CRC Press LLC
1 1 + x
tanh-1(x) = lnëÅ‚ öÅ‚ (S.51)
íÅ‚
2 1 - xłł
ëÅ‚
1 1 + 1 - x2 öÅ‚
sech-1(x) = lnìÅ‚ ÷Å‚ (S.52)
íÅ‚ Å‚Å‚
2 x
© 2001 by CRC Press LLC