Solved Examples and Worksheet for Finding Even or Odd Functions

Q1A function is said to be even if it's curve:

A. has symmetry with respect to x-axis and y-axis
B. has symmetry with respect to y-axis
C. has symmetry with respect to x-axis
D. cannot be determined

Solutions
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Step: 1
Function even means f(-x) = f(x).
Step: 2
It is possible the curve has symmetry with respect to y-axis.
Correct Answer is :   has symmetry with respect to y-axis
Q2If f (t) = sinh (6t), then which of the following is true?

A. Neither an even nor an odd function
B. An odd function
C. An even function
D. Not an odd function

Solutions
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Step: 1
f (t) = sinh (6t) = e6t-e- 6t2
  [Formula.]
Step: 2
f  (- t) = e- 6t-e- (- 6t)2
Step: 3
= e- 6t-e6t2
Step: 4
= - [(e6t-e- 6t)2]
Step: 5
= - f (t)
Step: 6
Since f  (- t) = - f  (t), the given function is an odd function.
Correct Answer is :   An odd function
Q3If f (t) = cosh (6t), then which of the following is true?
A. An even function
B. An odd function
C. Neither an even nor an odd function
D. Not an odd function

Solutions
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Step: 1
f  (t) = cosh (6t) = e6t+e- 6t2
  [Since, cosh (x) = ex+e-x2.]
Step: 2
f  (- t) = e- 6t+e- (- 6t)2
Step: 3
= (e6t+e- 6t)2 = f  (t)
Step: 4
Since f  (- t) = f  (t), the given function is an even function.
Correct Answer is :   An even function
Q4Which of the following is true for the function f(x) = sin 2x ?
I. f(x) is an odd function
II. f(x) is symmetric about origin
III. f(x) is symmetric about y - axis
IV. f(x) is not symmetric
V. f(- x) = f(x)
A. III only
B. III and IV
C. I and II
D. IV only
E. V only

Solutions
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Step: 1
f(x) = sin 2x
Step: 2
f(- x) = sin 2(- x) = - sin 2x
Step: 3
f(- x) = - f(x)
Step: 4
f(x) is an odd function.
  [If f(- x) = - f(x), then f(x) is an odd function.]
Step: 5
So, f(x) is symmetric about origin.
Correct Answer is :   I and II
Q5Which of the following is True for the function y = csc x ?

A. even function
B. odd function
C. odd and even function
D. can not be determined

Solutions
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Step: 1
If x is the measure of an angle with terminal side in quadrant I, then - x is the measure of an angle with terminal side in quadrant IV. Similarly, if x is the measure of a quadrant IV angle, then - x is the measure of a quadrant I angle. In either case, since csc x is negative in quadrant IV and positive in quadrant I, then csc(- x) = - csc x.
Step: 2
If x is the measure of an angle with terminal side in quadrant II, then - x is the measure of an angle with terminal side in quadrant III. Also, if x is the measure of a quadrant III angle, then - x is the measure of a quadrant II angle. In either case, since csc x is negative in quadrant III and positive in quadrant II. So, csc (- x) = - csc x
Step: 3
Since csc(- x) = - csc x for all values of x in the domain of the function, y = csc x is an odd function.
Correct Answer is :   odd function
Q6Find if g(x) = 2x3 - x - 3 an even function, an odd function, or neither.

A. odd function
B. even function
C. data insufficient
D. neither even nor odd

Solutions
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Step: 1
g(x) = 2x3 - x - 3
  [Given function.]
Step: 2
A function, f is an even function, if f(- x) = f(x)
Step: 3
g(- x) = 2(- x)3 - (- x) - 3 = - (2x3 - x + 3)
Step: 4
A function, f is an odd function, if f(- x) = - f(x)
Step: 5
- g(- x) = 2x3 - x + 3
Step: 6
g(x) ≠ g(- x) or g(x) ≠ - g(- x)
Step: 7
Thus the function g(x) is neither even nor odd.
Correct Answer is :   neither even nor odd
Q7Find whether the function f(x) = x8 + x2 - 5 an even function, or an odd function, or neither?

A. neither even nor odd
B. data insufficient
C. even function
D. odd function

Solutions
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Step: 1
f(x) = x8 + x2 - 5
Step: 2
f(- x) = (- x)8 + (- x)2 - 5 = x8 + x2 - 5
Step: 3
Since both f(x) and f(- x) are equal, the function is an even function.
Correct Answer is :   even function
Q8Is f(x) = 3x3 - 3x an even function, an odd function, or neither?

A. data insufficient
B. odd function
C. even function
D. neither even nor odd

Solutions
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Step: 1
f(x) = 3x3 - 3x
Step: 2
f(- x) = 3(- x)3 - (- 3x) = - (3x3 - 3x)
Step: 3
Since f(- x) = - f(x), the function is an odd function.
Correct Answer is :   odd function
Q9Is the function f(x) = e7x an even function, odd function, or neither?
A. odd function
B. data insufficient
C. neither even nor odd
D. even function

Solutions
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Step: 1
f(x) = e7x
Step: 2
A function, f is an even function, if f(- x) = f(x)
Step: 3
f(- x) = e- 7x
Step: 4
A function, f is an odd function, if f(- x) = - f(x)
Step: 5
- f(- x) = - e- 7x
Step: 6
Since f(x) ≠ f(- x) or f(x) ≠ - f(- x), the function f is neither even nor odd.
Correct Answer is :   neither even nor odd
Q10Find whether the function g(x) = e- 8x4an even function, odd function or neither.
A. data insufficient
B. neither even nor odd
C. odd function
D. even function

Solutions
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Step: 1
g(x) = e- 8x4
Step: 2
g(- x) = e- 8(- x)4 = e- 8x4
Step: 3
Since both g(x) and g(- x) are equal, the function is an even function.
Correct Answer is :   even function

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