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Relations and Functions
1. Relations
A relation is anything that can be represented by a set of ordered
pairs. The first elements, usually referred to as x-values for convenience,
come from a set called the domain. The second elements, the y-values, form
a set called the range.
Example:
Ex: {(0, 1), (1, – 2), (1, 3), (2, 7)} represents a relation.
Domain: {0,1,2}
Range: {1,-2,3,7}
- FUNCTIONS
A function is a relation (a set of ordered pairs) where each x-value
corresponds to a unique y-value.
2 main classifications
- Algebraic
a. Rational
b. irrational
- Transcendental
a. Elementary
b. Higher function
Functions are normally denoted by a single letter (like a variable).
The letters are typically, but not always, lower case.
Ex: The function f is defined by
2
f x = − x + 4 x + 5.
Notes:
The name of the function is f.
f x is pronounced “f of x.” It is a single quantity. It does
not mean f times x.
f x can be thought of as another name for y.
The same function could have been written
2
y = − x + 4 x + 5.
- The name of the independent variable is not important:
2
f x = − x + 4 x + 5
2
f t = − t + 4 t + 5
- For a given number k,
f ( k )
means the value of f (or y) when x = k.
- For a number k,
f ( x )= k
means the x-value(s) that make the value of
f be k (i.e., find x so that y = k)
- DOMAIN AND RANGE
DOMAIN: Set of all values in the independent variable (x - values).
RANGE: y values
- OPERATIONS ON FUNCTIONS
The following are definitions on the operations on functions.
a. The sum or difference of f and g, denoted by f ± g is the
function defined by (f ± g)(x) = f(x) ± g(x).
b. The product of f and g, denoted by f · g is the function defined by
(f·g)(x) = f(x)·g(x).
c. The quotient of f and g denoted by f/g is the function defined by
f(x)
g(x)
, where g(x) is not equal to zero.
d. The composite function of f and g denoted by f ο g is the function
defined by (f ο g)(x) = f(g(x)). Similarly, the composite function
of g by f, denoted by g ο f, is the function defined by
( g ο f)(x) = g(f(x)).
Examples:
- If f(x) = 2x + 1 and g(x) = 3x + 2, what
is (f+g)(x)?
Solution:
(f+g)(x) = f(x) + g(x)
= (2x + 1) + (3x + 2)
= 2x + 3x + 1 + 2
= 5x + 3
- What is (f • g)(x) if f(x) = 2x + 1 and g(x) = 3x + 2?
Solution:
(f • g)(x) = f(x) • g(x)
These are both the same function f. They have the same set of
ordered pairs and the same graphs.
THEOREMS ON LIMITS
THEOREM 1: The limit of the sum of two (or more) functions is equal to the sum of their
limits.
lim
x→a
[u(x) + v(x] = lim
x→a
u(x) + lim
x→a
v(x)
THEOREM 2: The limit of the product of two (or more) functions is equal to the product of
their limits.
lim
x→a
[u(x). v(x] = [lim
x→a
u(x) ] [lim
x→a
v(x) ]
THEOREM 3: The limit of the quotient of two functions is equal to the quotient of their
limits, provided the limit of the denominator is not zero.
lim
x→a
u(x)
v(x)
lim
x→a
u(x)
lim
x→a
v(x)
, if lim
x→a
v(x) ≠ 0
Solved Exercises
Evaluate the following limits.
- lim
x→ 4
(x
2
= lim
x→ 4
(x
2
) + lim
x→ 4
(3x) + lim
x→ 4
2
- lim
x→- 1
(2x
2
= lim
x→- 1
( 2 x
2
) + lim
x→- 1
(x) + lim
x→- 1
2
- lim
y→ 3
(y
3
= lim
y→ 3
y
3
− lim
y→ 3
y→ 3
3
- lim
y→- 2
(y
3
= lim
y→- 2
(y
3
) + lim
y→- 2
(5y) + lim
y→- 2
5. lim
x→- 0
2t
2
t
3
+3t - 4
lim
x→- 0
2t
2
lim
x→- 0
t
3
+3t - 4
2
3
CONTINUITY
A function f(x) is continuous at x= a if it satisfies all of
the following conditions:
- f(a) exists
- lim
x→a
f(x) exists
- lim
x→- 1
f(x) = f(a)
If not all conditions are satisfied, the function is
discontinuous at x=a.
Missing-point Discontinuity – occurs when condition 1 is not
satisfied. The curve appears continuous but actually the point
x=a is missing.
Finite Jumps – Occurs when the function has both left-hand and
right-hand limits which are different from each other.
Infinite Discontinuities – The curve approaches the line x=a
without ever touching it.
