LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

RELATION & FUNCTION - VERTICAL LINE TEST FOR A FUNCTION

__Vertical
Line Test For a Function –__

**You can test whether a given relation is a
function or not by applying the following tests ,**

**(i) Examine whether the first set, i.e., the
domain is fully used up or not **

**(ii) Examine whether the first members of all
the ordered pairs are different or not. In an arrow diagram, examine whether
each element of the first set has only one image in the second set or not.**

**(iii) The vertical line test. If it is possible
to draw a vertical line that intersects the graph of a relation in more than
one point, then the relation is not a function. If it is not possible to draw
such a vertical line, then the relation is a function.**

**It is not difficult to read the theory behind
this. If there are two points of the graph of a relation on a certain vertical
line, then the two corresponding ordered pairs in the relation have the same
first component but different second components. This means that the relation
is not a function by the definition of a function.**

**Example.1) Which of the following relations are
functions? Give reason. In case of a function, determine its domain and range.**

**(i) {(1, -2), (3, 7), (4, -6), (8, 11)}**

**(ii) {(1, 0), (1, -1), (2, 3), (4, 10)}**

**(iii) {(a, b), (b, c), (c, b), (d, c)}**

**Ans.) (i) and (iii) are functions because no
two of their ordered pairs have the same first components; (ii) is not a
function since the two ordered pairs (1, 0) and (1, -1) have the same first
components. **

**Domain of (i) is {1, 3, 4, 8}. Range is {-2, 7,
-6, 11}**

**Domain of (iii) is {a, b, c, d}. Range is {b,
c}**

**Example.2) Which of the following relations are
functions?**

**(i) y = 3x + 2**

**(ii) a is the capital of b where b ∈ B and B is the set of all countries, a ∈ A and A
is the set of capital cities of countries **

**(iii) y < x + 3**

**(iv) y is the maths teacher of x
where x represents any pupil taking up maths in a school **

**(v) y is a maths pupil of x, where x
represents any math teacher in a school.**

**Ans.) (i) Function. Each replacement of x gives one
and only one replacement of y which makes the sentence true.**

**(ii) Function. Because normally
each country has exactly one capital city.**

**(iii) Not a function. For each
replacement of x there are many replacements of y which makes the sentence
true.**

**(iv) Function. If we assumed
that every pupils only one maths teacher.**

**(v) Not a Function. Each math
teacher ordinarily has a number of pupils.**

**Example.3) State whether or not
each of the following diagrams defines a function of A = {a, b, c} into B = {x, y, z}**

**Ans.) (i) No, the domain set A is not entirely used
up.**

**(ii) No, two elements x & z
are assigned to the same element c in the domain set A. This will yield ordered
pairs (c, x), (c, z) having the same first component.**

**(iii) Yes, because a given
association is a function even if (a) one element of the second set B is
assigned to more than one element of the domain, (b) the second set B is not
entirely used up.**