# GMAT Math : Graphing a function

## Example Questions

### Example Question #1 : Graphing A Function

What is the domain of ?

all real numbers

all real numbers

Explanation:

The domain of the function specifies the values that  can take.  Here, is defined for every value of , so the domain is all real numbers.

### Example Question #3 : How To Graph A Function

What is the domain of ?

Explanation:

To find the domain, we need to decide which values  can take.  The  is under a square root sign, so  cannot be negative.   can, however, be 0, because we can take the square root of zero.  Therefore the domain is .

### Example Question #2 : Graphing A Function

What is the domain of the function ?

Explanation:

To find the domain, we must find the interval on which is defined.  We know that the expression under the radical must be positive or 0, so is defined when .  This occurs when and .  In interval notation, the domain is .

### Example Question #5 : How To Graph A Function

Define the functions  and  as follows:

What is the domain of the function  ?

Explanation:

The domain of  is the intersection of the domains of  and and  are each restricted to all values of  that allow the radicand  to be nonnegative - that is,

, or

Since the domains of  and  are the same, the domain of  is also the same. In interval form the domain of  is

### Example Question #6 : How To Graph A Function

Define

What is the natural domain of ?

Explanation:

The radical in and of itself does not restrict the domain, since every real number has a real cube root. However, since the expression  is in a denominator, it cannot be equal to zero, so the domain excludes the value(s) for which

27 is the only number excluded from the domain.

### Example Question #7 : How To Graph A Function

Define

What is the natural domain of  ?

Explanation:

Since the expression  is in a denominator, it cannot be equal to zero, so the domain excludes the value(s) for which . We solve for  by factoring the polynomial, which we can do as follows:

Replacing the question marks with integers whose product is  and whose sum is 3:

Therefore, the domain excludes these two values of .

### Example Question #8 : How To Graph A Function

Define .

What is the natural domain of ?

Explanation:

The only restriction on the domain of  is that the denominator cannot be 0. We set the denominator to 0 and solve for  to find the excluded values:

The domain is the set of all real numbers except those two - that is,

.