# GMAT Math : Functions/Series

## Example Questions

### Example Question #51 : Understanding Functions

Consider the function .

State whether this function is even, odd, or neither, and give the reason for your answer.

is odd because it is a polynomial of degree 3.

is odd because  for each value of  in the domain.

is not odd, because there exists at least one value of  for which  ;  is not even, because there exists at least one value of  for which .

is even because it is a polynomial of degree 3.

is even because  for each value of  in the domain.

is not odd, because there exists at least one value of  for which  ;  is not even, because there exists at least one value of  for which .

Explanation:

A function is odd if and only if  for each value of  in the domain; it is even if and only if  for each value of  in the domain. To disprove a function is odd or even, we need only find one value of  for which the appropriate statement fails to hold.

Consider :

, so  is not an odd function; , so  is not an even function.

### Example Question #51 : Understanding Functions

.

Evaluate .

Explanation:

First we evaluate . Since the parameter is negative, we use the first half of the definition of :

; since the parameter here is again negative, we use the first half of the definition of :

Therefore, .

### Example Question #1277 : Gmat Quantitative Reasoning

is defined to be the greatest integer less than or equal to .

Define  .

Evaluate .

Explanation:

### Example Question #191 : Algebra

If  and , what is ?

Explanation:

We start by finding g(2):

Then we find f(g(2)) which is f(4):

### Example Question #1279 : Gmat Quantitative Reasoning

Define two real-valued functions as follows:

Determine .

Explanation:

by definition.  is piecewise defined, with one defintion for negative values of the domain and one for nonnegative values. However,  is nonnegative for all real numbers, so the defintion for nonnegative numbers, , is the one that will always be used. Therefore,

for all values of .

### Example Question #51 : Functions/Series

Define two real-valued functions as follows:

Determine .

Explanation:

by definition.

is piecewise defined, with one defintion for negative values of the domain and one for nonnegative values.

If , then we use the definition . This happens if

or

Therefore, the defintion of   for  or  is

Subsquently, if , we use the defintion , since :

.

The correct choice is

### Example Question #51 : Understanding Functions

Define a function  on the real numbers as follows:

Give the range of the function.

Explanation:

This can be understood better by substituting , and, subsequently,  in the function's definition.

which is now in standard quadratic form in terms of .

Write this in vertex form by completing the square:

Substitute  back for , and the original function can be rewritten as

.

To find the range, note that . Therefore,

and

The range of  is the set .

### Example Question #51 : Understanding Functions

Define a function  on the real numbers as follows:

Give the range of the function.

Explanation:

This can be understood better by substituting , and, subsequently,  in the function's definition.

which is now in standard quadratic form in terms of .

Write this in vertex form by completing the square:

Substitute  back for . The original function can be rewritten as

can assume any real value; so, subsequently, can . But its square must be nonnegative, so

and

The range of  is

### Example Question #52 : Understanding Functions

If the functions and  intersect only when , and that , and , what is ?

Explanation:

In order to find exactly the  values where the equations intersect and when . We need to consider each piece of information seperately.

Let's start with . Plugging  into , we have . Plugging 0 into this, we have. This in turn equals 1, because we were given that piece of information in the beginning. So we end up with

Now let's shift our attention to "intersect only when " That means, if we plug 1 into both equations, we can set them equal to each other.

becomes becomes .

Now we have two different equations arising from the two previous paragraphs.

We can solve this system of equations using the substitution method.

Solving for  in the first equation gives .

Plugging this equation in for  the 2nd equation gives . Using algebra on this equation we get

Now we are ready to use the quadratic formula to solve for .

Finally, since we're told in the beginning that , we must pick the plus sign in our solution for . Hence

.

### Example Question #51 : Understanding Functions

Define two real-valued functions as follows:

Determine .