### All GMAT Math Resources

## Example Questions

### Example Question #1 : Graphing A Function

What is the domain of ?

**Possible Answers:**

all real numbers

**Correct answer:**

all real numbers

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 #2 : Graphing A Function

What is the domain of ?

**Possible Answers:**

**Correct answer:**

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 #3 : Graphing A Function

What is the domain of the function ?

**Possible Answers:**

**Correct answer:**

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 #4 : Graphing A Function

Define the functions and as follows:

What is the domain of the function ?

**Possible Answers:**

**Correct answer:**

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 #5 : Graphing A Function

Define

What is the natural domain of ?

**Possible Answers:**

**Correct answer:**

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 #6 : Graphing A Function

Define

What is the natural domain of ?

**Possible Answers:**

**Correct answer:**

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 #7 : Graphing A Function

Define .

What is the natural domain of ?

**Possible Answers:**

**Correct answer:**

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,

.

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