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f ( x ) = ( x ) 2 = x 2 = f ( x ) .

In contrast, looking at [link] again, if a function f is symmetric about the origin, then whenever the point ( x , y ) is on the graph, the point ( x , y ) is also on the graph. In other words, f ( x ) = f ( x ) . If f has this property, we say f is an odd function, which has symmetry about the origin. For example, f ( x ) = x 3 is odd because

f ( x ) = ( x ) 3 = x 3 = f ( x ) .

Definition

If f ( x ) = f ( x ) for all x in the domain of f , then f is an even function    . An even function is symmetric about the y -axis.

If f ( x ) = f ( x ) for all x in the domain of f , then f is an odd function    . An odd function is symmetric about the origin.

Even and odd functions

Determine whether each of the following functions is even, odd, or neither.

  1. f ( x ) = −5 x 4 + 7 x 2 2
  2. f ( x ) = 2 x 5 4 x + 5
  3. f ( x ) = 3 x x 2 + 1

To determine whether a function is even or odd, we evaluate f ( x ) and compare it to f ( x ) and f ( x ) .

  1. f ( x ) = −5 ( x ) 4 + 7 ( x ) 2 2 = −5 x 4 + 7 x 2 2 = f ( x ) . Therefore, f is even.
  2. f ( x ) = 2 ( x ) 5 4 ( x ) + 5 = −2 x 5 + 4 x + 5 . Now, f ( x ) f ( x ) . Furthermore, noting that f ( x ) = −2 x 5 + 4 x 5 , we see that f ( x ) f ( x ) . Therefore, f is neither even nor odd.
  3. f ( x ) = 3 ( x ) / ( ( x ) 2 + 1 } = −3 x / ( x 2 + 1 ) = [ 3 x / ( x 2 + 1 ) ] = f ( x ) . Therefore, f is odd.
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Determine whether f ( x ) = 4 x 3 5 x is even, odd, or neither.

f ( x ) is odd.

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One symmetric function that arises frequently is the absolute value function    , written as | x | . The absolute value function is defined as

f ( x ) = { x , x < 0 x , x 0 .

Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if x < 0 , then | x | = x > 0 , and if x > 0 , then | x | = x > 0 . However, for x = 0 , | x | = 0 . Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if x = 0 , the output | x | = 0 . We conclude that the range of the absolute value function is { y | y 0 } . In [link] , we see that the absolute value function is symmetric about the y -axis and is therefore an even function.

An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -4 to 4. The graph is of the function “f(x) = absolute value of x”. The graph starts at the point (-3, 3) and decreases in a straight line until it hits the origin. Then the graph increases in a straight line until it hits the point (3, 3).
The graph of f ( x ) = | x | is symmetric about the y -axis.

Working with the absolute value function

Find the domain and range of the function f ( x ) = 2 | x 3 | + 4 .

Since the absolute value function is defined for all real numbers, the domain of this function is ( , ) . Since | x 3 | 0 for all x , the function f ( x ) = 2 | x 3 | + 4 4 . Therefore, the range is, at most, the set { y | y 4 } . To see that the range is, in fact, this whole set, we need to show that for y 4 there exists a real number x such that

2 | x 3 | + 4 = y .

A real number x satisfies this equation as long as

| x 3 | = 1 2 ( y 4 ) .

Since y 4 , we know y 4 0 , and thus the right-hand side of the equation is nonnegative, so it is possible that there is a solution. Furthermore,

| x 3 | = { ( x 3 ) if x < 3 x 3 if x 3 .

Therefore, we see there are two solutions:

x = ± 1 2 ( y 4 ) + 3 .

The range of this function is { y | y 4 } .

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For the function f ( x ) = | x + 2 | 4 , find the domain and range.

Domain = ( , ) , range = { y | y −4 } .

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Key concepts

  • A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.
  • If no domain is stated for a function y = f ( x ) , the domain is considered to be the set of all real numbers x for which the function is defined.
  • When sketching the graph of a function f , each vertical line may intersect the graph, at most, once.
  • A function may have any number of zeros, but it has, at most, one y -intercept.
  • To define the composition g f , the range of f must be contained in the domain of g .
  • Even functions are symmetric about the y -axis whereas odd functions are symmetric about the origin.

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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