A function is odd if and only if, for every in its domain, ; it is even if and only if, for every in its domain, .

We see that and . Therefore, , so is false for at least one . cannot be even.

For a function to be odd, since , it follows that ; since is its own opposite, must be 0. However, ; cannot be odd.

The correct choice is neither.

A function is odd if and only if, for all , ; it is even if and only if, for all , . Therefore, to answer this question, determine by substituting for , and compare it to both and .

, so is not even.

, so is not odd.

A function is even if, for each in its domain,

.

It is odd if, for each in its domain,

.

Substitute for in the definition:

Since , is an even function.

, by definition, is an odd function if, for all in its domain,

, or, equivalently

One implication of this is that for to be odd, it must hold that . If , then, since

for nonnegative values, then, by substitution,

This condition is satisfied.

Now, if is negative, is positive. it must hold that

,

so for all

,

the correct response.

A function is even if and only if, for all in its domain, . It follows that if , then

.

No restriction is placed on any other value as a result of this information, so the answer is false.

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

Also, it can be seen to be symmetrical about the origin. Consequently, for each in the domain, - the function is odd.

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

Also, it is seen to be symmetric about the origin. Consequently, for each in the domain, - the function is odd.

A function is even if, for each in its domain,

.

It is odd if, for each in its domain,

.

Substitute for in the definition of :

Since is even, , so

This makes an odd function.

Symmetry across the origin is symmetry across .

Determine the inverse of the function. Swap the x and y variables, and solve for y.

Subtract 3 on both sides.

Divide by negative two on both sides.

The answer is:

, by definition, is an odd function if, for all in its domain,

, or, equivalently

One implication of this is that for to be odd, it must hold that . Since is explicitly defined to be equal to 0 here, this condition is satisfied.

Now, if is negative, is positive. it must hold that

,

so for all

This is the correct choice.