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Finding the Domain and Range of a Function (1 of 2)


The domain of a function is the set of all allowable values of the independent variable, commonly known as the x-values. To find the domain, I need to identify particular values of x that can cause the function to "misbehave" and exclude them as valid inputs to the function.

The values of x that can result to the following conditions are not included in the domain of the function.

Not allowed to happen Why?
getting a denominator of zero division of zero is undefined
having to get the square root of negative number square root of negative numbers is imaginary


The range of a function is the set of output values when all x-values in the domain are evaluated into the function, commonly known as the y-values. This means I need to find the domain first in order to describe the range.

To find the range is a bit trickier than finding the domain. I highly recommend that you use a graphing calculator to have an accurate picture of the function. However, if you don't have one, I encourage you to sketch some of the basic functions by hand. Either way, it is crucial that you have a good idea of how the graph looks like in order to correctly describe the range of the function.

Direction: Find the domain and range of the following functions.
1) linear function: y=3x-1 See solution
2) quadratic function: y=x^2+3 See solution
3) quadratic function that opens downward: y=-x^2+2 See solution
4) y=x^2+4x-1 See solution


Example 1: Find the domain and range of the function linear equation y=3x-1.

The first thing I've observed is that there is no square root symbol or denominator in this problem. This is wonderful because getting a square root of a negative number or a division of zero is not possible with this function. Since there are no x-values that can make the function to output invalid results, I can easily claim that the domain is all x values. However, it is much better to write it in set notation or interval notation.

Here's the summary of the domain and range of the given function written in two ways...

  Domain Range
Set Notation {x|x ∈ℜ } {y|y∈ℜ}
Interval Notation (−∞,+∞) (−∞,+∞)


Because the function involved is a line, I can predict that the range is all y values . It can definitely go as high or as low without any limits. Look at the graph below to understand what I mean.

It's always wonderful to see graph of the function together with its domain and range, in pictorial format.

example 1 domain and range  graph where domain is all x-values, and  range is all y-values


Example 2: Find the domain and range of the function quadratic equation y=x^2+3.

I can see that I can plug any values of x into the function and it will produce a valid output. So, I can safely say that its domain is all x values. This time, however, I need to be careful how to describe the range. Is it going to be all y values? Well, I don't think so, because I know this function is a parabola and one of its traits is having a high point (maximum) or a low point (minimum). To be safe, I will first graph it.

example 2  domain and range  graph where domain is all x-values, while range is y values greater than or equal to 3

The graph of the parabola has a low point at y = 3 and it can go as high as it wants. Using inequality, I will write the range as > 3.

Summary of domain and range in tabular form:

  Domain Range
Set Notation domain  in set notation form: {x|x ∈ℜ } range in set notation: {y|y∈ℜ, y≥3}
Interval Notation (−∞,+∞) range of example 2 interval notation: [3,+∞)


Example 3: Find the domain and range of the function quadratic equation y=-x^2+2.

I hope that the previous example has given you the idea on how to work this out. This is a quadratic function, thus, the graph will be parabolic. I know that this will also have either a minimum or a maximum. Since the coefficient of the x2 term is negative, the parabola opens downward and therefore has a maximum (high point). The domain should be all x values because there are no values that when substituted to the function will yield "bad results".

Although the range is easy to find, I'd rather "play safe" and graph it again.


example 3 domain and range  graph where domain is all x-values, and range is less than or equal to 2

The parabola has a maximum value at y = 2 and it can go down as low as it wants. The range is simply y < 2.


The summary of domain and range is the following:

  Domain Range
Set Notation example 3,  domain in set notation: {x|x ∈ℜ } example 3, range in set notation: {y|y∈ℜ, y≤2}
Interval Notation example 3, domain in interval (−∞,+∞) example 3, range in interval notation: (-∞,2]



Example 4: Find the domain and range of the function quadratic equation y = x^2 + 4x - 1.

Just like our previous examples, a quadratic function will always have a domain of all x values.

I want to go over this particular example because the minimum or maximum is not quite obvious. Notice though that the parabola is in the Standard Form, y = ax2 + bx + c.

I want to transform this into the Vertex Form, y = a (x-h)2 + k, where vertex is (h,k) using the method of Completing the Squares.

y=x^2+4x-1, then rewrite as y = (x+2)^2-5

The parabola opens upward and the vertex must be a minimum. The coordinate of the vertex is...

(h,k) = (-2,-5) where the lowest value in y is k=-5

I can now see that this parabola has a minimum value at y = −5, and can go up to positive infinity.

The range should be y > 5.


To verify it using its graph, I have this diagram.

example 4 domain and range  graph where the domain is all x values, and range is y≥-5


Our summary of domain and range is...

  Domain Range
Set Notation example 4, domain in set notation:  {x|x ∈ℜ} range in set notation: {y|y∈ℜ, y≥-5}
Interval Notation example 4, domain in interval notation: (−∞,+∞) range in interval notation is [-5, +∞)


Go to the next page to see more examples...


Practice Problems with Answers
Worksheet 1 Worksheet 2


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