HOW TO TELL IF A FUNCTION IS EVEN OR ODD

Determine whether each function is even, odd, or neither.

Problem 1 :

 g(x) = x³ - x

Solution :

This function is odd because

g(-x) = (-x)³ - (-x)

= -x³ + x

= -(x³ - x)

= -g(x)

The function g(x) is odd.

Problem 2 :

h(x) = x² + 1

Solution :

This function is even because

h(-x) = (-x)² + 1

= x² + 1

= h(x)

The function h(x) is even.

Problem 3 :

 f(x) = x³ - 1

Solution :

Substituting -x for x produces

f(-x) = (-x)³ - 1

= -x³ - 1

= - (x³ + 1)

You can conclude that

f(-x) ≠ f(x)

and

f(-x) ≠ -f(x)

So, the function is neither even nor odd.

Determine whether the function is even, odd, or neither. Then describe the symmetry.

Problem 4 :

   f(x) = 5 - 3x

Solution :

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = 5 - 3(-x)

f(-x) = 5 + 3x

f(-x) cannot be expressed as either as f(x) or -f(x).

So, f(x) is neither even nor odd function.

Problem 5 :

g(x) = x⁴ - x² - 1

Solution :

To know g(x) is odd or even function, substitute -x for x in g(x).

Then, we have

g(-x) = (-x)⁴ - (-x)² - 1

g(-x) = x⁴ - x² - 1

g(-x) = g(x)

So, g(x) is even function.

Problem 6 :

h(x) = 2x³ + 3x

Solution :

To know h(x) is odd or even function, substitute -x for x in h(x).

Then, we have

h(-x) = 2(-x)³ + 3(-x)

h(-x) = -2x³ - 3x

h(-x) = -h(x)

So, h(x) is odd function.

Problem 7 :

f(t) = t² + 2t - 3

Solution :

To know f(t) is odd or even function, substitute -t for t in f(t).

Then, we have

f(-t) = (-t)² + 2(-t) - 3

f(-t) = t² - 2t - 3

f(-t) cannot be expressed as either as f(t) or -f(t).

So, f(t) is neither even nor odd function.

Problem 8 :

g(x) = x³ - 5x

Solution :

To know g(x) is odd or even function, substitute -x for x in g(x).

Then, we have

g(-x) = (-x)³ - 5(-x)

g(-x) = -x³ + 5x

g(-x) = -g(x)

So, g(x) is odd function.

Problem 9 :

f(x) = x√1 - x²

Solution :

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = (-x)√1 - (-x)²

f(-x) = -x√1 - x²

f(-x) = -(x√1 - x²)

f(-x) = -f(x)

So, f(x) is odd function.

Problem 10 :

f(x) = x√x + 5

Solution :

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = -x√-x + 5

f(-x) cannot be expressed as either as f(x) or -f(x).

So, f(x) is neither even nor odd function.