Even and Odd Functions Video Lecture | Sets and Functions - JEE

even functions and odd functions a
function y equals f of X is an even
function of X if f of negative x equals
f of X or it is an odd function of X if
f of negative x equals negative of f of
X for every X in the functions domain if
the function doesn't satisfy either of
the above properties then it is neither
an even function nor an odd function
the names even and odd come from powers
of X if Y is an even power of X as in y
equals x square and y equals x raised to
the power 4 it is an even function of X
because square of negative x equals x
square and negative x raised to the
power 4 equals x raised to the power 4
if y is an odd power of x as in y equals
x and y equals x cubed it is an odd
function of X because the first power of
negative x equals negative of x and cube
of negative x equals negative of x cube
of an even function is symmetric about
the y-axis this is because in case of an
even function the value of y is same for
X a negative x the graph of an even
function does not change if we reflect
it across y axis
raph of an odd function is symmetric
about the origin this is because the
value of y at negative x is the negative
of the value of y at x the graph of an
odd function does not change if we
rotate it by 180 degrees about the
origin let's rotate the graph by 180
degrees clockwise now let's rotate the
graph by 180 degrees anti-clockwise
clearly the graph remains unchanged in
both the cases consider the function f
of x equals x squared plus 1 let's see
whether the given function is even odd
or neither to check whether the given
function is odd or even first we need to
determine F of negative x for that we
will substitute negative x in place of x
and f of X doing this we get F of
negative x equals square of negative x
plus 1 square of negative x equals x
square therefore f of negative x equals
x square plus 1 which equals f of X
since f of negative x equals f of X the
given function is even it can be seen
that the graph of this function is
symmetric about the y-axis
now consider the function f of X equals
x plus 1 pause this video and try to
figure out whether the given function is
even odd or neither as discussed earlier
first we need to determine f of negative
X which in this case equals negative x
plus 1 now negative of f of x equals
negative of x plus 1 which equals
negative x minus 1 clearly F of negative
x neither equals f of X nor it equals
negative of f of X therefore the given
function is neither even nor odd it can
be seen that the graph of this function
is neither symmetric about the y-axis
nor it is symmetric about the origin
now consider the function f of X equals
x cubed plus X again pause this video
and try to figure out whether the given
function is even odd or neither let's
determine F of negative x it equals cube
of negative x plus negative x cube of
negative x equals negative X cube so f
of negative x equals negative x cubed
minus x we can factor out the negative
sign doing this we get F of negative x
equals negative of X cubed plus X which
equals negative of f of X since f of
negative x equals negative of f of X the
given function is odd it can be seen
that the graph of this function is
symmetric about the origin
now consider the function f of X equals
zero again pause this video and try to
figure out whether the given function is
even odd or neither since f of X is a
constant function therefore F of
negative x also equals zero since f of X
equals zero negative of f of X also
equals 0 therefore f of negative x
equals f of X as well as negative of f
of X so the given function is even as
well as odd the graph of this function
lies along the x axis and is symmetric
about the origin as well as the y-axis
in fact f of x equals 0 is the only real
function that is both an even function
and an odd function