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:  

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. 

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. 

To find vertical asymptotes, factor each quadratic and then simplify.

The vertical asymptotes are the zeros of the denominator, so x=3 is a vertical asymptote.

Horizontal asymptotes are found by analyzing the coefficients of the first term in each equation. The line  (where a is the coefficient of the first term in the numerator and b is the coefficient of the first term in the denominator) is the horizontal asymptote. So we have y=1 as the horizontal asymptote.

The vertical asymptotes of a rational function are always the zeroes of the polynomial in the denominator. So to solve this problem all we need to do is find the zeroes of .

We can use the quadratic formula, , for a polynomial . In this case, a=4, b=-8, and c=-5. So we must plug these values into the quadratic formula:

And we get the two roots, .