**Finding the Domain and Range of a Function **

Let’s start with the domain of a function…

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.

Now, how about the range of a function?

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.

**Example 1:** Find the domain and range of the function

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…

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 2:** Find the domain and range of the function

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.

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 **y ≥ 3**.

Summary of domain and range in tabular form:

**Example 3:** Find the domain and range of the function

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 x^{2} 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.

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:

**Example 4:** Find the domain and range of the function

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 = ax^{2} + 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.

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

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 5:** Find the domain and range of the function

Remember that I can’t have x-values which can result to having a negative number under the square root symbol. To find the domain (“good values of x”), I know that it is allowable to take the square root of either zero , or any positive number. My plan is to find all the values of x satisfying that condition. It will become the domain itself.

I would let the expression under the radical, x-2, greater than or equal to zero; and then solve the inequality. Check out my other lesson on how to solve inequalities.

This radical function has a domain of **x ≥ 2**. I need to be careful finding the range of this function. The graph of the function looks like this…

The radical function starts at y = 0 , and can go as high as it wants (positive infinity). You may think that this function grows slowly (slow increase in y values) thus can’t reach extremely large values. However, you must consider that plugging in sufficiently large values of x ( i.e. in billions of trillions) can result to a very large output values of y.

Therefore, I will claim that the **range** of this function is **y ≥ 0**.

This is the summary of the domain and range written both in set and interval notations.

**Example 6:** Find the domain and range of the function

The acceptable values under the square root are zero and positive numbers. So I will let the “stuff” inside the radical equal or greater than zero, and then solve for the required inequality.

Now, the domain of the function is **x ≤ 5**. Just like in our previous examples, I will graph the function to determine the range.

The radical function starts at y = 0, and then slow but steadily decreasing in values all the way down to negative infinity. This makes the range **y ≤ 0**. Below is the summary of both domain and range.

**Example 7:** Find the domain and range of the function

This function contains a denominator. This tells me that I must find the x-values that can make the denominator zero to prevent the** undefined case** to happen.

Here, our domain is **all x-values but does not include x = 2**. It makes a lot sense because I can plug any values of x into the function with the exception of x = 2, and the function will have valid outputs. The graph below shows that x = 2 is actually a vertical asymptote (see dashed orange line).

To find the range is a bit tricky. Looking at the graph carefully, I see that it goes up without any limit and goes down without any limit as well. However, I won’t rush to claim that the range is all y values. There is something going on as the graph moves to the right without bound. Do you see that it gets closer and closer to zero? Similarly, this characteristic is also happening as the graph moves to the left without bound. It also gets very close to zero but not quite. This quick analysis gives me the intuition that maybe **y cannot equal zero**.

Doing some “common sense” analysis to show that y cannot equal zero (y ≠ 0).

Going back to the original function…

If I want y to equal zero, I need to find values of x to do the job. If you think about it, there are no x values that can make it happen. Why? Because in order for the rational expression to become zero, the NUMERATOR MUST BE ZERO. But the numerator is not zero, in fact, it is 5!

This tells me that I could never find an input (domain) to have an output of zero (range).

Therefore, the range is **all y-values but does not include y = 0**. The open circle in the graph below denotes that y = 0 is **excluded** from the range.

This is our final summary for the domain and range of the given rational function.

**Example 8:** Find the domain and range of the function

The domain of this function is exactly the same as in Example 7. The idea again is to **exclude** the values of x that can make the denominator zero. Obviously, that value is x = 2 and so the **domain is all x values except x = 2**.

To find the range, I will heavily depend on the graph itself. It is possible to sketch it by hand using more advanced graphing techniques but I will leave it for another lesson. Anyway, the graph shows that it covers all possible y-values: goes up and down without bounds, and no breaks in between. Therefore the **range is all y values**.

The domain and range written in two ways…