y = [x] - x find even and odd function.
Answers
Answer:
You may be asked to "determine algebraically" whether a function is even or odd. To do this, you take the function and plug –x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f (–x) = f (x), so all of the signs are the same), then the function is even.
Explanation:
To test if the fuction is an even function, you would replace
x
with
−
x
, and see if the resulting equation is the same as the original equation.
First, write out the original equation as a function by replacing
y
with
f
(
x
)
:
f
(
x
)
=
x
+
x
2
Second, replace
x
with
−
x
:
f
(
−
x
)
=
−
x
+
(
−
x
)
2
Third, simplify the equation:
f
(
−
x
)
=
−
x
+
x
2
Finally, compare it to the original equation:
f
(
x
)
=
x
+
x
2
≠
f
(
−
x
)
=
−
x
+
x
2
Since
f
(
x
)
≠
f
(
−
x
)
, the function is not even.
To test if the fuction is an odd function, you would test to see if
f
(
−
x
)
=
−
f
(
x
)
. If the two equations are the same, then the function is odd.
First, write out the original equation as a function by replacing
y
with
f
(
x
)
:
f
(
x
)
=
x
+
x
2
Second, multiply both sides of the equation by
−
1
:
−
f
(
x
)
=
−
1
(
x
+
x
2
)
Third, simplify the equation:
−
f
(
x
)
=
−
x
−
x
2
Finally, compare it to the equation from our even function test:
f
(
−
x
)
=
−
x
+
x
2
≠
−
f
(
x
)
=
−
x
−
x
2
Since
f
(
−
x
)
≠
−
f
(
x
)
, the function is not odd.
If the function is not odd or even, then it is neither.