What is an Even Function? Definition and Examples

What is an even function? A function f is an even function if f(-x) = f(x) for all x in the domain of f.

For example f(x) = x2 is an even function

f(-4) = (-4)2 = -4 × -4 = 16 and f(4) = (4)=  4 × 4 = 16

f(-4) = f(4) = 16

f(-3) = (-3)2 =  = -3 × -3 = 9 and f(3) = (3)= 3 × 3 = 9

f(-3) = f(3) = 9

f(-1) = (-1)2 = -1 × -1 = 1 and f(1) = (1)=  1 × 1 = 1

f(-1) = f(1) = 1

In general, let x represent any number in the domain of f.

Then, f(-x) = (-x)2 = -x × -x = x2 and f(x) = x2

f(-x) = f(x) = x2

For f(-4) = f(4) = 16, the points are (-4, 16) and (4, 16)

For f(-3) = f(3) = 9, the points are (-3, 9) and (3, 9)

For f(-1) = f(1) = 1, the points are (-1, 1) and (1, 1)

Notice that f(0) = 02 = 0, so there is a single point (0,0)

Now, try to graph these seven points on a coordinate system.

(-4, 16), (4, 16), (-3, 9), (3, 9), (-1, 1), (1, 1), and (0, 0)

From the graph above, we can make the following two important observations:

• If a point (-x,y) is on the graph, the point (x,y) is also on the graph.
• The function is symmetric with respect to the y-axis

More examples of even functions

• f(x) = x2 - 5
• f(x) = x4 - 3x2
• f(x) = |x|
• f(x) = |x| + 6
• f(x) = x6
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