Introduction to Functions - Algebra II

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.

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 .

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:

In order for the function to be real, the value inside of the square root must be greater than or equal to zero. The domain refers to the possible values of the independent variable (x-value) that allow this to be true.

For this term to be real, must be greater than or equal to zero.

The domain of a rational function is the set of all values of for which the denominator is not equal to 0 (the value of the numerator is irrelevant), so we set the denominator to 0 and solve for to find the excluded values.

This is a quadratic function, so we factor the expression as , replacing the question marks with two numbers whose product is 8 and whose sum is . These numbers are , so

becomes

So either

, in which case ,

or , in which case .

Therefore, 2 and 4 are the only numbers excluded from the domain.

The domain of a rational function is the set of all values of for which the denominator is not equal to 0, so we set the denominator to 0 and solve for .

This is a quadratic function, so we factor the expression as , replacing the question marks with two numbers whose product is 9 and whose sum is . These numbers are , so

becomes

,

or .

This means that , or .

Therefore, 3 is the only number excluded from the 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.

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:

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 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).

This function does not have any such restrictions; any value of will result in a real number. The domain is thus unlimited, ranging from negative infinity to infinity.

Domain:

This function represents a parabola that has been shifted 15 units to the left and 2 units up from its standard position.

The vertex of a standard parabola is at (0,0). By shifting the graph as described, the new vertex is at (-15,2). The value of the vertex represents the minimum of the range; since the parabola opens upward, the maximum will be infinity. Note that the range is inclusive of 2, so you must use a bracket "\[".

Minimum: 2 (inclusive), maximum: infinity

Range:

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:

The range of a function is the group of corresponding values for a given domain ( values). Plug each value into the function to find the range:

The range is .

All inputs are valid. There is nothing you can put in for x that won't work.

All inputs are valid. There is nothing you can put in for x that won't work.

All inputs are valid. There is nothing you can put in for x that won't work.

All inputs are valid. There is nothing you can put in for x that won't work.