To check for symmetry, we are going to do three tests, which involve substitution. First one will be to check symmetry along the x-axis, replace .

This isn't equivilant to the first equation, so it's not symmetric along the x-axis.

Next is to substitute .

This is not the same, so it is not symmetric along the y-axis.

For the last test we will substitute , and 

This isn't the same as the orginal equation, so it is not symmetric along the origin.

The answer is it is not symmetric along any axis.

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 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.

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.

, 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.

The relation passes the Vertical Line test, as seen in the diagram below, in that no vertical line can be drawn that intersects the graph than once:

An function is odd if and only if its graph is symmetric about the origin, and even if and only if its graph is symmetric about the -axis. From the diagram, we see neither is the case.

, 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.

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: