Range and Domain

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.

A basic knowledge of absolute value and its functions is valuable for this problem. However, if you do not know what the typical shape of an absoluate value function looks like, one can always plug in values and plot points.

Upon doing so, we learn that the -values (domain) are not restricted on either end of the function, creating a domain of negative infinity to postive infinity.

If we plug in -100000 for , we get 100000 for .

If we plug in 100000 for , we get 100000 for .

Additionally, if we plug in any value for , we will see that we always get a real, defined value for .

**Extra Note: Due to the absolute value notation, the negative (-) next to the is not important, in that it will always be made positive by the absolute value, making this function the same as . If the negative (-) was outside of the absolute value, this would flip the function, making all corresponding -values negative. However, this knowledge is most important for range, rather than domain.

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.

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:

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 .

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 refers to all the possible x-values that can be existent on the given function. Do not confuse this with the range, since this represents all the existent y-values on the graph.

Since there are no discontinuities for any x-value that we may substitute, the domain is all real numbers.

The answer is:

The domain of a function refers to the viable value inputs. Common domain restrictions involve radicals (which cannot be negative) and fractions (which cannot have a zero denominator). Both of these restrictions can be found in the given function.

Let's start with the radical, which must be greater than or equal to zero:

Next, we will look at the fraction denominator, which cannot equal zero:

Our final answer will be the union of the two sets.

Minimum: 2 (inclusive), maximum: infinity

Exclusion: 22

Domain:

There are two limitations in the function: the radical and the denominator term. A radical cannot have a negative term, and a denominator cannot be equal to zero. Based on the first restriction (the radical), our term must be greater than or equal to zero. Based on the second restriction (the denominator), our term cannot be equal to 4. Our final answer will be the union of these two sets.

Minimum: 0 (inclusive), maximum: infinity

Exclusion: 4

Domain:

The natural domain of the composite function is defined to be the intersection two sets.

One set is the natural domain of . Since is a polynomial, its domain is the set of all real numbers.

The other set is the set of all values of such that that is in the domain of . Since the radicand of the square root in must be nonnegative,

, and , the domain of

Therefore, the other set is the set of all such that

Substitute:

This holds for all real numbers, so this set is also the set of all real numbers.

The natural domain of is the set of all real numbers.