# Precalculus : Relations and Functions

## Example Questions

### Example Question #41 : Relations And Functions

What is the domain of the function given by:

Explanation:

cos(x) is definded for all reals. cos(x) is always between -1 and 1. Thus  . The value inside the square is always positive. Therefore the domain is the set of all real numbers.

### Example Question #42 : Relations And Functions

Find the domain of

Explanation:

Since

for all real numbers, the denominator is never 0 .Therefore the domain is the set

of all real numbers.

### Example Question #43 : Relations And Functions

http://physiologyweb.com/calculators/nernst_potential_calculator.html

The Nernst equation is very important in physiology, useful for measuring an ion's potential across cellular membranes. Suppose we are finding an ion's potential of a potassium ion at body temperature. Then the equation becomes:

Where  is the ion's electrical potential in miniVolts and  is a ratio of concentration.

What is the domain and range of ?

Explanation:

Apart from the multiplication by , this function is very similar to the function

The logarithmic function has domain x>0, meaning for every value x>0, the function  has an output (and for x = 0 or below, there are no values for log x)

The range indicates all the values that can be outputs of the . When you draw the graph of y = log(x), you can see that the function extends from -infinity (near x = 0), and then extends out infinitely in the positive x direction.

Therefore the Domain:

and the is Range:

### Example Question #44 : Relations And Functions

Find the domain and range of the given function.

Domain: All real numbers

Range:

Domain: All real numbers

Range: All real numbers

Domain:

Range:

Domain:

Range: All real numbers

Domain: All real numbers

Range:

Explanation:

The domain is the set of x-values for which the function is defined.

The range is the set of y-values for which the function is defined.

Because the values for x can be any number in the reals,

and the values for y are never negative,

Domain: All real numbers

Range:

### Example Question #45 : Relations And Functions

Find the domain and range of the function.

Domain: All real numbers

Range: All real numbers

Domain: All real numbers

Range:

Domain:

Range:

Domain:

Range: All real numbers

Domain:

Range:

Explanation:

The domain is the set of x-values for which the function is defined.

The range is the set of y-values for which the function is defined.

Because the values for x are never negative,

and the values for y are never negative,

Domain:

Range:

### Example Question #46 : Relations And Functions

What is the domain of

Explanation:

As long as the number under the square root sign is greater than or equal to , then the corresponding x-value is in the domain. So to figure out our domain, it is easiest to look at the equation and determine what is NOT in the domain. We do this by solving  and we get . We now look at values greater than and less than , and we can see that when , the number under the square root will be negative. When , the number will be greater than or equal to . Therefore, our domain is anything greater than or equal to 6, or .

### Example Question #47 : Relations And Functions

What is the range of

Explanation:

Because the only term in the equation containing an  is squared, we know that its value will range from  (when ) to  (as  approaches ). When  is large, a constant such as  does not matter, but when  is at its smallest, it does. We can see that when  will be at its minimum of . This number gets bracket notation because there is an  value such that .

### Example Question #48 : Relations And Functions

What is the domain of the function?

Explanation:

Any value can be inputed in the exponetial.

### Example Question #49 : Relations And Functions

Find the domain of the function:

Explanation:

The square cannot house any negative term or can the denominator be zero. So the lower limit is  since  cannot be , but any value greater than it is ok. And the upper limit is infinity.

### Example Question #50 : Relations And Functions

What is the domain for the function?