GMAT Functions with Examples

The GMAT is not an easy exam. It is obvious that an exam designed to assess your ability to handle the rigors of an MBA or EMBA program would not be a walk in the park! And it’s a well-known fact that most students consider the Quantitative Reasoning section to be the most challenging of the three sections on the GMAT. Furthermore, there is no doubt that students rate GMAT Functions to be in the top 3 of the most difficult of the 20 topics tested on the exam.

In this article, we’ll take a deep dive into the world of GMAT functions. We’ll cover basic concepts, such as the definition of a function and the domain and range of a function. Then we’ll look at various function types, ranging from quadratic functions to exponential functions, and we’ll solve many GMAT-quality example questions. We’ll take you on a whirlwind tour of all you need to know to become a functional functions master!

Here are the topics we’ll cover:

• What Is a Function?
  • Function Notation
    • Example 1: Evaluate a Function
• Domain and Range of a Function
  • Example 2: Domain of a Function
• Quadratic Functions
  • Minimum and Maximum Values of a Quadratic Function
    • Example 3: Minimum Value of a Function
    • Example 4: Maximum Value of a Function
• Composite Functions
  • Example 5: Composite Functions
• Exponential Functions
  • Example 6: Exponential Functions
• Symbolic Functions
  • Example 7: Symbolic Functions
• Summary
• What’s Next?

Let’s begin by discussing the definition of a function.

What Is a Function?

In simplest terms, a function is like a machine that takes an input and gives an output. For example, if the input is 5 and the machine instruction is to square the input, then the output is 5^2, or 25. Symbolically, we would summarize this mathematical operation as:

f(x) = x^2    For any input x, the function instruction is to square x.

f(5) = 5^2    When x = 5, the function squares it

f(5) = 25    The output of the function when x = 5 is 25.

Here are some other examples of functions.

f(x) = 4x – 17             f(x) = √x              f(m) = (m^3 + 3) / 2m – 3      

KEY FACT:

A function takes an input and produces an output.

Next, let’s discuss function notation.

Function Notation

The function notation presented above may be unfamiliar, so let’s look at it a bit more. First, f(x) is pronounced “f of x.” It is simply saying that the value of our function is based on the input value x. Note that f(x) does not denote multiplication. It is, again, a notation that identifies a function.

We can use letters other than x to represent the input value. Earlier, we saw a function defined as f(m), and m acts identically to x.

Even though we usually use the letter “f’” to indicate a function (because it is the first letter of the word “function”), we may see functions expressed as g(x) or h(x).

KEY FACT:

The standard notation denoting a function is f(x), which is pronounced “f of x.” Other letters can also be used.

Let’s look at an example.

Let’s now discuss the domain and range of a function.

Domain and Range of a Function

The set of all inputs that a function can use is called the domain of the function, and the set of all numbers that are outputs of the function is called the range of the function.

KEY FACT:

A function’s domain is the set of all inputs, and its range is the set of all outputs.

For example, if we have the function f(x) = 3x, we can let x be any real number; that is, there are no restrictions on what value x can take on. However, if we have the function f(x) = 1 / x, we see that x is not allowed to be 0 because division by 0 is not defined. Thus, for the function f(x) = 1 / x, we say that the domain is the set of all real numbers except 0.

Another domain restriction is illustrated by the function f(x) = √x. Here, we see that x cannot take on any value less than 0. The domain—the set of allowable values for x—can be described as “any nonnegative number.”

The two situations described above are examples of the two “domain restrictions” we must be concerned with for GMAT purposes.

KEY FACT:

For GMAT purposes, there are two domain restrictions: (1) We cannot divide by 0, and (2) we cannot take the square root of a negative number.

The GMAT tests us on some specific types of functions, the first of which is quadratic functions. Let’s discuss.

Quadratic Functions

A quadratic function is one whose shape is a parabola when it is graphed in the xy coordinate plane. You may recall that an equation of the form y = ax^2 + bx + c yields a parabola.

If the value of a is positive, the parabola is up-opening, and if the value of a is negative, the parabola is down-opening. Similarly, the function f(x) = ax^2 + bx + c is algebraically identical, and the result is also a parabola. And, for our purposes, we can consider these two equations to be interchangeable.

KEY FACT:

A function of the form f(x) = ax^2 + bx + c is a parabola. If the value of a is positive, the parabola opens up, and if the value of a is negative, the parabola opens down.

Minimum and Maximum Values of a Quadratic Function

On the GMAT, you may be asked to determine the minimum or maximum value of a quadratic function if you are given an algebra question about functions or if you are given a word problem that models a quadratic function.

To determine either the minimum or maximum output of a function f(x) = ax^2 + bx + c, we can use the following:

If a > 0, the minimum output of the quadratic function occurs when we input x = (-b) / 2a

If a < 0, the maximum output of the quadratic function occurs when we input x = (-b) / 2a

We must remember that once we determine our x value, we must plug it back into the original function to determine the minimum or maximum output.

KEY FACT:

For a function f(x) = ax^2 + bx + c, if a > 0 the function has a minimum value, and if a < 0, the function has a maximum value.

Let’s look at an example.

Let’s now solve a word problem involving the maximum value of a function.

Composite Functions

Composite functions, also called compound functions, are functions that are nested into other functions. You may see them expressed as g(f(x)) or f(g(x)) or f(f(x)), for example. The key to understanding a composite function is that the output of one function is the input of the next function. Let’s look at an illustration.

If f(x) = 3x + 4 and g(x) = x^2 – 5, what is the value of g((f(2)) ?

We work from the inside out, so we first evaluate f(2):

f(2) = 3(2) + 4 = 6 + 4 = 10

The value of f(2) is 10, so now we evaluate the g function:

g(f(2)) = g(10) = 10^2 – 5 = 100 – 5 = 95

KEY FACT:

A composite function is a nested function such that the output of the inner function becomes the input of the outer function.

Let’s practice with an example.

Exponential Functions

You may have heard the term “exponential growth,” and this concept is often tested on the GMAT. The function that describes exponential growth is given by f(x) = ab^x.

Note that in an exponential function, the variable appears as an exponent. In this equation, a represents the function’s initial value, while the constant b represents the constant multiplier that we will multiply a number of times to get this initial value of a. The variable x represents the period of time over which the growth occurs.

KEY FACT:

The exponential function is of the form f(x) = ab^x, where a is the initial value of the function, b is the constant multiplier, and x is the amount of time over which growth occurs.

Consider the following mini-example:

There are 200 bacteria in a colony, and they are undergoing exponential growth. The number of bacteria doubles every hour. How many bacteria are there at the end of 3 hours?

First, we see that the initial number of bacteria is 200, so a = 200. Next, we see that the number of bacteria doubles every hour, so the value of b is 2.

Thus, the exponential growth function for this bacteria colony is f(x) = 200 (2^x)

Finally, since we want to determine the population of bacteria after 3 hours, we can plug in 3 for x:

f(x) = 200 (2^3) = 1,600

Let’s answer a GMAT example question about exponential growth.

Symbolic Functions

Some GMAT function questions will present us with symbols representing a particular mathematical operation or sequence of operations.

For example, we may encounter something such as x @ y = x^2 + 3y  or x # y = 8x – 2y. These are called symbolic functions. Note that the symbols linking x and y do not denote a particular operation; they simply indicate a relationship between x and y.

As strange as these questions may seem, they define functions, and we can follow their rules just like we would with a function such as f(x) = 3x – 5.

For example:

If x @ y = x^2 – xy, what is 3 @ 2?

This equation tells us that the @ symbol indicates an operation for which we square x and subtract the product of x and y. So, to calculate 3 @ 2, we can let x = 3 and y = 2 and plug these values into the formula x^2 – xy:

3 @ 2 = 3^2 + (3)(2) = 9 + 6 = 15

KEY FACT:

A symbolic function inserts a symbol, such as @ or #, between two variables to indicate a relationship between the two variables, which is given in the expression that follows.

Consider the following example.

Summary

In this article, we have covered the basics of functions on the GMAT. Let’s look at the highlights.

• A function is a rule that uses an input to create an output. The notation f(x) indicates a function.
• The domain of a function is the set of all possible inputs, and the range is the set of all possible outputs.
• For GMAT purposes, there are two domain restrictions: (1) we cannot divide by 0 and (2) we cannot take the square root of a negative number.
• A composite function is a nested function in which the output of one function acts as the input for the next function. The notation g(f(x)), for example, indicates that the output value of f(x) becomes the input value for g(x).
• An exponential function models growth that increases by a multiplicative factor during each time period. The standard form for exponential growth is f(x) = ab^x, where a is the initial value, b is the growth factor, and x is the number of time periods over which exponential growth occurs.
• Symbolic functions show a mathematical relationship between two variables. For example, the symbolic function x # y = 4x – 5y^2 defines a relationship between two variables x and y.

What’s Next?

We have covered just one of the 20 math topics on the GMAT. If you would like to see the big picture of GMAT math, read our article about the structure and breakdown of the Quantitative Section.

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