### All GMAT Math Resources

## Example Questions

### Example Question #71 : Graphing

Let be the point of intersection of the graphs of these two equations:

Evaluate .

**Possible Answers:**

The system has no solution.

**Correct answer:**

Substitute and for and , respectively, and solve the resulting system of linear equations:

Multiply the first equation by 2, and the second by 3, on both sides, then add:

Back-solve:

We need to find both and to ensure a solution exists. By substituting back:

and

We check this solution in both equations:

- true.

- true.

is the solution, and , the correct choice.

### Example Question #71 : Graphing

The graph of function has vertical asymptote . Which of the following *could* give a definition of ?

**Possible Answers:**

**Correct answer:**

Given the function , the vertical asymptote can be found by observing that a logarithm cannot be taken of a number that is not positive. Therefore, it must hold that , or, equivalently, and that the graph of will never cross the vertical line . That makes the vertical asymptote, so it follows that the graph with vertical asymptote will have in the position. The only choice that meets this criterion is

### Example Question #11 : Graphing

The graph of a function has -intercept . Which of the following *could* be the definition of ?

**Possible Answers:**

All of the other choices are correct.

**Correct answer:**

All of the other choices are correct.

All of the functions are of the form . To find the -intercept of such a function, we can set and solve for :

Since we are looking for a function whose graph has -intercept , the equation here becomes , and we can examine each of the functions by finding the value of .

:

:

:

:

All four choices fit the criterion.

### Example Question #1061 : Gmat Quantitative Reasoning

The graph of a function has -intercept . Which of the following *could* be the definition of ?

**Possible Answers:**

**Correct answer:**

All of the functions take the form

for some integer . To find the choice that has -intercept , set and , and solve for :

In exponential form:

The correct choice is .

### Example Question #15 : Graphing

Define a function as follows:

Give the -intercept of the graph of .

**Possible Answers:**

**Correct answer:**

Set and evaluate to find the -coordinate of the -intercept.

Rewrite in exponential form:

.

The -intercept is .

### Example Question #16 : Graphing

Define functions and as follows:

Give the -coordinate of a point at which the graphs of the functions intersect.

**Possible Answers:**

The graphs of and do not intersect.

**Correct answer:**

The graphs of and do not intersect.

Since , the definition of can be rewritten as follows:

Since , the definition of can be rewritten as follows:

First, we need to find the -coordinate of the point at which the graphs of and meet by setting

Since the common logarithms of the two polynomials are equal, we can set the polynomials themselves equal, then solve:

However, if we evaluate , the expression becomes

,

which is undefined, since a negative number cannot have a logarithm.

Consequently, the two graphs do not intersect.

### Example Question #17 : Graphing

The graph of a function has -intercept . Which of the following *could* be the definition of ?

**Possible Answers:**

None of the other responses gives a correct answer.

**Correct answer:**

All of the functions are of the form . To find the -intercept of a function , we can set and solve for :

.

Since we are looking for a function whose graph has -intercept , the equation here becomes , and we can examine each of the functions by finding the value of and seeing which case yields this result.

:

:

:

:

The graph of has -intercept and is the correct choice.

### Example Question #18 : Graphing

Define a function as follows:

A line passes through the - and -intercepts of the graph of . Give the equation of the line.

**Possible Answers:**

**Correct answer:**

The -intercept of the graph of can befound by setting and solving for :

Rewritten in exponential form:

The -intercept of the graph of is .

The -intercept of the graph of can be found by evaluating

The -intercept of the graph of is .

If and are the - and -intercepts, respectively, of a line, the slope of the line is . Substituting and , this is

.

Setting and in the slope-intercept form of the equation of a line:

### Example Question #19 : Graphing

Define functions and as follows:

Give the -coordinate of a point at which the graphs of the functions intersect.

**Possible Answers:**

The graphs of and do not intersect.

**Correct answer:**

Since , the definition of can be rewritten as follows:

.

Find the -coordinate of the point at which the graphs of and meet by setting

Since the common logarithms of the two polynomials are equal, we can set the polynomials themselves equal, then solve:

The quadradic trinomial can be "reverse-FOILed" by noting that 2 and 6 have product 12 and sum 8:

Either , in which case

or

, in which case

Note, however, that we can eliminate as a possible -value, since

,

an undefined quantity since negative numbers do not have logarithms.

Since

and

,

is the correct -value, and is the correct -value.