How to find the domain of a function - Algebra 1

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.

and stands for OR in Set Builder Notation

From the figure one can see the Range varies from to .

The expression under the radical is defined for all real values of since the index of the radical is 3.

Since the expression under the radical must be greater than or equal to zero, hence when , the . Thereafter the is an increasing function.

Every real number has a real cube root, so the radical does not restrict the domain of . The denominator of the expression does restrict the domain, however, in that it cannot be equal to 0. This happens only if:

or, equivalently, . Therefore, 1000 is the only real number not in the domain of .

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 relation is a function if and only if no -coordinate is paired with more than one -coordinate. We test each of these four values of to see if this happens.

:

The points become:

Since -coordinate 1 is paired with two -coordinates, 2 and 9, the relation is not a function.

:

The points become:

Since -coordinate 3 is paired with two -coordinates, 0 and 9, the relation is not a function.

:

The points become:

Since -coordinates 3 and 5 are each paired with two different -coordinates, the relation is not a function.

:

The points become:

Since each -coordinate is paired with one and only one -coordinate, the relation is a function. is the correct choice.

Finding the domain is like finding out what possible values you can plug in without getting an error message on your calculator. We are given the function:

To find the domain, first, know that you can't take the square root of a negative number. It turns out that the minimum number you can take the square root of is zero. Any number above zero, everything is good. Therefore, can equal 0 or above. This is written as an inequality as:

Solve for by subtracting 1 on both sides

So that's our domain!

The graph will open to the left. The contents inside the square root cannot be negative.

Set the inside equal to zero.

Any number greater than one-third will be invalid, but any number below will be the domain of this function.

The domain is:

At , there is a hole in this function:

At all other values of x, both positive and negative, this function will be defined.

To find the range, we first need to find the domain of the function. Then we will determine the range by finding the output values based on the domain.

First, we can factor out x from the denominator:

Since the function will be undefined when the denominator equals 0, we know that x=0 and x=1 are not in the domain of this function. We also know that this function is undefined for all negative number since the function includes a root of an even degree. The exponent under the square root does not change this, since cubing a negative number will still result in a negative number, as with any odd degreed exponent.

So, this function is defined at all non-negative values except for 0 and 1.

The domain is defined as the input values or x values of a set.

So domain: 4,3,7

The range is defined as the output values or y values of a set.

So range: 2,7,3

In order for the set to be a function, each input value must have only one corresponding output value. In this example, the input value 4 has two output values: 2 and 3. The set is not a function.

The domain is defined as the input values or x values of a set. We can see that x is never less than zero

So domain: 0 ≤ x < ∞

The range is defined as the output values or y values of a set. We can see that we reach all values of y

So range: y = all real numbers

In order for the set to be a function, each input value must have only one corresponding output value. Another way of interpreting this graphically is that the graph must be able to pass the vertical line test. If we draw a vertical line on this graph, it corsses the parabola twice. This means there are multiple y output values for a single x input value.

Therefore, the graph is not a function

The domain of a set of ordered pairs is the values.

The values are the first number in each set of coordinates.

The values are:

The domain consists of all values that the input can be without making the output unreasonable. In our problem, the only condition that would dissatisfy the equations parameters is a negative inside the square root. However, having inside the square root makes this a bit tricky, because we have to consider that squaring this value will always yield something positive. Thus, we cannot have any values of whose squares are strictly less than . Thus, the domain must be all values of that are greater than or equal to and less than or equal to .