# Precalculus : Relations and Functions

## Example Questions

### Example Question #51 : Relations And Functions

What is the domain of the function below?

Explanation:

The denomiator factors out to:

The denominator becomes zero when . But the function can exist at any other value.

### Example Question #52 : Relations And Functions

What is the domain of the function below?

Explanation:

Cannot have a negative inside the square root. The value of  has to be  for the inside of the square root to be at least . This is the lower bound of the domain. Any value of  greater than  exists.

### Example Question #53 : Relations And Functions

Explanation:

The natural log function does not exist if the inside value is negatuve or zero. The points where the inside becomes negative are  or . If  is greater than , both terms,  and , are positive. If  is less than , both terms are negative and multiply to become positive. If the  value is between  and , only one term will be negative and result in a , which does not exist.

### Example Question #54 : Relations And Functions

What is the domain of the function?

Explanation:

The value inside a natural log function cannot be negative or . At , the inside is  and any  value less than  cannot be included, because result will be a negative number inside the natural log.

### Example Question #55 : Relations And Functions

What is the domain of the function?

Does not exist anywhere.

Explanation:

Exponentials cannot have negatives on the inside. However, the expoential will convert any  value into a positive value.

### Example Question #56 : Relations And Functions

What is the domain of the function?

Explanation:

Looking at the denominator, the function cannot exist at . The natural log function cannot have a  or negative inside. Since the  value is raised to the power of , any negative  value will be convert to a positive value. However, the function will not exist if the inside of the natural is , where .  will exist any where else.

### Example Question #57 : Relations And Functions

What is the domain of the function?

Explanation:

The denominator becomes  where , and the inside of the natural log also becomes  at . The function will not exist at these two points.  The  value cannot be less than , becuase that will leave a negative value inside in the natural log.

### Example Question #58 : Relations And Functions

Find the domain of the function.

Explanation:

Simplify:

Even though the  cancels out from the numerator and denominator, there is still a hole where the function discontinues at . The function also does not exist at , where the denominator becomes

### Example Question #1 : Determine If A Relation Is A Function

Which of the following expressions is not a function?

Explanation:

Recall that an expression is only a function if it passes the vertical line test. Test this by graphing each function and looking for one which fails the vertical line test. (The vertical line test consists of drawing a vertical line through the graph of an expression. If the vertical line crosses the graph of the expression more than once, the expression is not a function.)

Functions can only have one y value for every x value. The only choice that reflects this is:

### Example Question #2 : Determine If A Relation Is A Function

Suppose we have the relation  on the set of real numbers  whenever . Which of the following is true.

The relation is a function because  holds and  also holds.

The relation is not a function because  holds but  does not.

The relation is a function because for every , there is only one  such that  holds.

The relation is a function because every relation is a function, since that's how relations are defined.

The relation is not a function because  and  both hold.

The relation is not a function because  and  both hold.

Explanation:

The relation is not a function because  and  hold. If it were a function,  would hold only for one . But we know it holds for  because  and . Thus, the relation  on the set of real numbers  is not a function.