In the **previous post**, we have learned how to analyze equations of functions and determine their domain and range. We have observed that the range of the functions and are the set of **real numbers** greater than or equal to since squaring a number or getting its absolute value results to or a positive real number. We also learned the value under the square root sign must be greater than or equal to 0. Lastly, we have learned that we cannot **divide by zero** because it will make the function **undefined**.

In this post, we are going to continue our discussion by examining functions with equations more complicated than those in the second part of this series.

**Squares and Absolute Values**

1.

*Domain*: The function is defined for any real number , so the domain of is the set of real numbers.

*Range*: The minimum value of is for any real number and . So, the minimum value of the function is . We can make the value of the function as large as possible by increasing the absolute value of . So, the range of the function is the set of real numbers greater than or equal to or in interval notation.

2.

*Domain:* The function is defined for any real number , so the domain of is the set of real numbers.

*Range:* The minimum value of any number inside the absolute value sign is (here, that happens at ). So adding to will make the minimum value of the range . So the range is the set of real numbers greater than or equal to or in interval notation.

**Division by 0**

3.

*Domain*: The denominator of the equation of the function must not equal to , so we find the value that will make it . That is, we equate to . Now, gives us . Therefore, must not equal to . So, the domain of the function is the set of real numbers except or . Notice, that to get the restrictions in the domain we equate the denominator to and solve for . The value of is the restriction in the domain.

*Range*: Just like the function in Part 2, we can have any value for , but it’s impossible for it to be no matter how small is. So, the range is the set of real numbers except 0 or

4.

*Domain:* The denominator of the equation of the function must not equal to , so we find the value that will make it . That is, we equate to . Now, means . Therefore, must not equal . So, the domain of the function is the set of real numbers except or .

*Range*: Left as an exercise

**Radicals**

5.

*Domain:* We have learned that the value inside the radical must be positive or 0, so,

Squaring both sides,

.

So, the domain of the function is the set of real numbers greater than or equal to 5. That is in interval notation.

*Range:* The value inside the radical sign must be greater than or equal to 0, so the range is the set of real numbers greater than or equal to 0.

6.

*Domain:*

Squaring both sides,

.

So, the domain is the set of real numbers greater than or equal to -7/2 or in interval notation.

*Range: *We know that , so the range is the set of real numbers greater than or equal to 0 or in interval notation.

In the **next post**, we are going to summarize what we have learned in the three posts.