EVEN AND ODD FUNCTIONS
About "Even and odd functions"
Even and odd functions :
Let f(x) be a function.
To find whether f(x) is even or odd, we have to replace "x" by "-x" in f(x)
Now, let us plug x = -x in f(x) and find whether f(x) is odd or even.
f(-x) = f(x) -----> f(x) is even function
f(-x) = - f(x) -----> f(x) is odd function
Even and odd functions - Application
Even though we have many applications of even and odd functions, let us consider the important application of even and odd functions in integral calculus.
If f(x) is even function,
If f(x) is odd function,
Even and odd functions - Practice problems
Problem 1 :
Let f(x) = x³, is f(x) odd or even function ?
Solution :
To know f(x) is odd or even function, let us plug x = -x in f(x).
Then, we have
f(-x) = (-x)³
f(-x) = -x³
f(-x) = - f(x)
Hence, f(x) is odd function
Problem 2 :
Let f(x) = x² + 2, is f(x) odd or even function ?
Solution :
To know f(x) is odd or even function, let us plug x = -x in f(x).
Then, we have
f(-x) = (-x)² + 2
f(-x) = x² + 2
f(-x) = f(x)
Hence, f(x) is even function.
Problem 3 :
Let f(x) = x³ + 5, is f(x) odd or even function ?
Solution :
To know f(x) is odd or even function, let us plug x = -x in f(x).
Then, we have
f(-x) = (-x)³ + 5
f(-x) = -x³ + 5
Here we can not define f(-x) in terms of f(x).
Hence, f(x) is neither odd nor even function.
Important note :
In trigonometric ratios, if we have negative angle, we have to understand that the angle will fall in the IV th quadrant.
In IV th quadrant, the trigonometric ratios "cos" and "sec" are positive and all other trigonometric ratios are negative
Problem 4 :
Is sinx odd or even function ?
Solution :
Let f(x) = sinx
To know f(x) is odd or even function, let us plug x = -x in f(x).
Then, we have
f(-x) = sin(-x)
Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant "sin" is negative.
So, we have
f(-x) = - sinx
f(-x) = - f(x)
f(x) is odd function
Hence, sinx is odd function.
Problem 5 :
Is cscx odd or even function ?
Solution :
Let f(x) = cscx
To know f(x) is odd or even function, let us plug x = -x in f(x).
Then, we have
f(-x) = csc(-x)
Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant "csc" is negative.
So, we have
f(-x) = - cscx
f(-x) = - f(x)
f(x) is odd function
Hence, cscx is odd function.
Problem 6 :
Is secx odd or even function ?
Solution :
Let f(x) = secx
To know f(x) is odd or even function, let us plug x = -x in f(x).
Then, we have
f(-x) = sec(-x)
Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant "sec" is positive.
So, we have
f(-x) = secx
f(-x) = f(x)
f(x) is even function
Hence, secx is even function.
Problem 7 :
Is cosx odd or even function ?
Solution :
To know f(x) is odd or even function, let us plug x = -x in f(x).
Then, we have
f(-x) = cos(-x)
Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant "cos" is positive.
So, we have
f(-x) = cosx
f(-x) = f(x)
f(x) is even function
Hence, cosx is even function.
Problem 8 :
Is tanx odd or even function ?
Solution :
Let f(x) = tanx
To know f(x) is odd or even function, let us plug x = -x in f(x).
Then, we have
f(-x) = tan(-x)
Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant "cos" is negative.
So, we have
f(-x) = - tanx
f(-x) = - f(x)
f(x) is odd function
Hence, tanx is odd function.
Problem 9 :
Let f(x) = sinx + tanx, is f(x) odd or even function ?
Solution :
To know f(x) is odd or even function, let us plug x = -x in f(x).
Then, we have
f(-x) = sin (-x) + tan(-x)
Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant both "sin" and "tan" are negative.
So, we have
f(-x) = - sinx - tanx
f(-x) = - (sinx + tanx)
f(-x) = - f(x)
Hence, f(x) is odd function.
Note : The sum or difference of two odd functions is always odd.
Problem 10 :
Let f(x) = secx + cosx, is f(x) odd or even function ?
Solution :
To know f(x) is odd or even function, let us plug x = -x in f(x).
Then, we have
f(-x) = sec(-x) + cos(-x)
Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant both "sec" and "cos" are positive.
So, we have
f(-x) = secx + cosx
f(-x) = f(x)
Hence, f(x) is even function.
Note : The sum or difference of two even functions is always even.
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