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 can see that 

Of course, 

.

Therefore,  is even by definition.

To find an oblique asymptote of a rational function whose numerator has higher degree than its denominator, as is the case here, divide the former by the latter, as follows:

Divide the leading term of the dividend by that of the divisor:

Place this in the quotient, and multiply this by the divisor:

Subtract this from the dividend. The figure should look like this:

Repeat with the difference:

The figure now looks like this:

The difference has degree less than that of the divisor, so the division is finished. The oblique asymptote is the quotient, 

The domain includes the values that go into a function (the x-values) and the range are the values that come out (the  or y-values). A sine curve represent a wave the repeats at a regular frequency. Based upon this graph, the maximum  is equal to 1, while the minimum is equal to –1. The x-values span all real numbers, as there is no limit to the input fo a sine function. The domain of the function is all real numbers and the range is .

A function has to pass the vertical line test, which means that a vertical line can only cross the function one time.  To put it another way, for any given value of , there can only be one value of .  For the function , there is one value for two possible  values.  For instance, if , then .  But if , as well.  This function fails the vertical line test.  The other functions listed are a line,, the top half of a right facing parabola, , a cubic equation, , and a semicircle, . These will all pass the vertical line test.

The domain is the set of possible value for the variable. We can find the impossible values of by setting the denominator of the fractional function equal to zero, as this would yield an impossible equation.

Now we can solve for .

There is no real value of that will fit this equation; any real value squared will be a positive number.

The radicand is always positive, and is defined for all real values of . This makes the domain of  the set of all real numbers.

To find the domain, find all areas of the number line where the fraction is defined.

because the denominator of a fraction must be nonzero.

Factor by finding two numbers that sum to -2 and multiply to 1.  These numbers are -1 and -1.

The domain is the set of x-values that make the function defined.

This function is defined everywhere except at , since division by zero is undefined.

Using  as the input () value for this equation generates an output () value that contradicts the stated condition of .

Therefore  is not a valid value for  and not in the equation's domain:

This function is a parabola that has been shifted up five units. The standard parabola has a range that goes from 0 (inclusive) to positive infinity. If the vertex has been moved up by 5, this means that its minimum has been shifted up by five. The first term is inclusive, which means you need a "[" for the beginning.

Minimum: 5 inclusive, maximum: infinity

Range: 

The domain represents the acceptable  values for this function. Based on the members of the function, the only limit that you have is the non-allowance of a negative number (because of the square root). The square and the linear terms are fine with any numbers. You cannot have any negative values, otherwise the square root will not be a real number.

Minimum: 0 inclusive, maximum: infinity

Domain: