# GMAT Math : Functions/Series

## Example Questions

### Example Question #161 : Algebra

Which of these functions is an example of a function with an inverse?

Explanation:

For a function  to have an inverse, if , then . We can show this is not the case for four of these functions by providing one counterexample in each case.

However, we can show that  has an inverse function by demonstrating that if , then .

### Example Question #21 : Understanding Functions

Define

Which of the following would be a valid alternative way of expressing the definition of ?

Explanation:

If , then  ,and subsequently,

If , then  ,and subsequently,

### Example Question #23 : Understanding Functions

Which of these functions is neither an even function nor an odd function?

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. We can identify each of these functions as even, odd, or neither by evaluating each at  and comparing the resulting expression to the definition of the function.

making  even.

making  odd.

making  odd.

making  even.

Since neither   nor  is neither even nor odd, making this the correct choice.

### Example Question #22 : Understanding Functions

If , then what does  equal?

Explanation:

This problem can be evaluated by determining what the value of the parentheses is and then using that to evaluate the rest of the term.

The term then reduces to

### Example Question #21 : Understanding Functions

Find the next term in the series

Explanation:

We can see that we have a geometric series.  The geometric factor can be found by dividing the second term by the first.  Doing this, we get   To find the next term in the series, we simply multiply the last term by the geometric factor to get .

### Example Question #23 : Understanding Functions

Solve for .

Explanation:

First, solve for .

Next, solve for .

### Example Question #24 : Understanding Functions

Define an operation  as follows:

if either  or , but not both, are negative, and  otherwise.

Evaluate:

Explanation:

Evaluate each of  and   separately. Since in both cases, the numbers have the same sign, replace the numbers for  and  in the second expression - that is, simply add them.

### Example Question #25 : Understanding Functions

Define an operation  as follows:

For all real numbers ,

if , and  if

Evaluate:

Explanation:

Evaluate  and  separately.

Since , use

for

Since , use

for

### Example Question #26 : Functions/Series

Define the operation  as follows:

For real

if both  and  are negative, and  otherwise.

Evaluate

Explanation:

Evaluate each of  and  separately. Since in both cases, at least one number is positive, replace the numbers for  and  in the second expression.

### Example Question #25 : Understanding Functions

Which of the following would be a valid alternative definition of the function

?

Explanation:

If , then  and  are both positive, so

If , then , then  is positive and  is negative, so

If , then  and  are both negative, so