EVEN AND ODD FUNCTIONS GRAPHS AND TABLES

A Function can be classified as Even, Odd or Neither. This classification can be determined graphically or algebraically.

How to check if the graph is odd ?

The graph will be symmetric with respect to the origin.

In other words :

If you spin the picture upside down about the Origin, the graph looks the same!

How to check if the graph is even ?

The graph will be symmetric with respect to the y-axis.

Problem 1 :

Which function is an even function?

a. y = h(x) = √x            b. y = p(x) = x³

c. y = R(x) = 1/x           d. y = |x|

Solution :

Option a :

y = h(x) = √x

Put x = -x

h(-x) = √-x

The function is not defined for negative values

Option b :

y = p(x) = x³

Put x = -x

p(-x) = (-x)³

p(-x) = -x³

It is odd function.

Option c :

y = R(x) = 1/x

Put x = -x

R(-x) = 1/(-x)  ==>  -1/x

R(-x) = -1/x

It is odd function.

Option d :

y = |x|

Put x = -x

y = |-x|

y = x

that is, f(-x) = f(x). It is even function.

Problem 2 :

Examine these functions. Which statement is correct?

a. Graph I is the graph of an even function.

b. Graph II is symmetric to the x-axis.

c. Graph III is symmetric to the y-axis.

d. Graph IV is an odd function.

Solution :

Option a :

Graph I is the graph of an even function.

Option b :

Graph II is symmetric to the x-axis.

Here the y-axis is like a mirror.

Option c :

It is symmetric about origin. But option c says, it is symmetric to the y-axis. So, it is incorrect.

Option d :

It is symmetric about origin. So, option d is correct.

Problem 3 : 

Which function could be an even function?

a) A function known as d(x) where d(3) = 10 and d (-3) = -10

b) A function known as k(x) where k(7) = 20 and k(-7) = 20

c) A function known as M(x) where M(4) = 10 and M (-4) = 0

d) A function known as C(x) where C(7) = - 6 and C(-7) = 6

Solution :

Even function will be symmetric about y-axis.

All points that lie on the curve will be on the rule given below,

(x, y) ==> (-x, y)

Option a :

d(3) = 10 and d (-3) = -10

When x = 3, y = 10

when x = -3, y = -10

It does not satisfy the rule above. So, it is not symmetric about y-axis and it is not even.

Option b :

k(7) = 20 and k(-7) = 20

When x = 7, y = 20

when x = -7, y = 20

It satisfies the above rule. So, it is even function.

Problem 4 :

Start with the incomplete graph of p(x) as shown. Complete the graph if p(x) is symmetric to the origin

Solution :

Even function will be symmetric about origin.

All points lie on the curve will follow the rule given below.

(x, y) ==> (-x, -y)

Problem 5 :

Start with the incomplete graph of f(x) as shown. Complete the graph if f(x) is symmetric to the y-axis.

Solution :

Even function will be symmetric about y-axis.

All points lie on the curve will follow the rule given below.

(x, y) ==> (-x, y)

Problem 6 :

If g(x) is an odd function and g(5) = -7, which one of the following must be true?

a. g(-5) = 7      b. g(-5) = -7     c. g(-7) = 5     d. g(7) = - 5

Solution :

Odd function is symmetric about origin. The points that lie on the curve will follow the rule below.

(x, y) ==> (-x, -y)

If g(5) = -7, the g(-5) = 7

Option a is correct.

Problem 8 :

Which function is not even (symmetric to the y-axis)?

a. y = -4 x² + 3        b. y = 2 |x| - 6

c. y = x² +6x + 5           d. y = 6

Solution :

If the function satisfies the condition, f(-x) = f(x), then it is even function.

Option a :

f(x) = -4 x² + 3 

f(-x) = -4 (-x)² + 3 

f(-x) = -4 x² + 3 

f(-x) = f(x)

It is even

Option b :

f(x) = 2 |x| - 6

Put x = -x

f(-x) = 2|-x| - 6

f(-x) = 2|x| - 6

f(-x) = f(x)

So, it is even.

Option c :

f(x) = x² +6x + 5 

f(-x) = (-x)² + 6(-x) + 5 

f(-x) = x² - 6x + 5 

f(-x) ≠ f(x)

It is not even. So, option c is correct.