GRE Exam: A Simple Guide To Algebra - Formulas, Functions, and Sequences

The next type of algebra question concerns situations where the question gives you a formula to determine some value.

Formulas
In formula questions, the question will define a formula for you and will ask you to substitute values or variables into the formula to arrive at an answer.

A typical question would be:
A company estimates the number of potential customers, P, in a day by using the following formula: , where c = the number of customers the company had the previous day and z = the number of competitors for that month.

If the company had 10 customers yesterday and has 2 competitors this month, how many customers will the company have today?

SOLUTION: Substitute the given values into the formula. Substitute 10 for c and 2 for .
 

Formulas with Unknown Amounts
Sometimes, formula questions will not give you values for the variables. Instead, they will ask you to determine how changes in certain variables affect the output of the formula. In these cases, you should always plug in numbers! For example:
The formula for the volume of a sphere is .

Example: If the radius of a sphere is doubled, the volume of the resulting sphere will be how many times the volume of the original sphere?
2
4
8
16
32

SOLUTION: Let the original radius = 2. The original volume is thus .

After the radius doubles, the new volume is . .

The correct answer is C.

Functions
You can think of a function as a recipe. Recipes specify certain steps you need to take to achieve a desired dish; in the same way, functions specify certain operations you need to perform to achieve a given outcome. Because of their strange notation, functions tend to intimidate a lot of people, but it is important to remember that fundamentally functions are just substitution!

A simple function question would be the following:
If, for all numbers, x, f(x) is defined by f(x) = 3x + 5, then f(7) =

It is important to note that “f(x)” does not mean f × x. f is the name of the function, and the term in the parentheses represents the input of the function.

Using the recipe analogy, this formula tells you that to get the value (also known as “output”) of the function, f, you must multiply what is inside of the parentheses by 3 and then add 5.

So in the preceding example, to get the value of f(7), you must multiply 7 by 3 and then add 5. 3(7) + 5 = 26.

Going from Output to Input
In the preceding example, you were given the formula for the function and its input, and you were asked to arrive at the output. Sometimes you will be given the formula for the function and its output, and you will be asked to arrive at the input. Look at the following example:

For all positive numbers, x, the function f is defined as f(x) = (x + 3)2 – 5. If f(a) = 59, then a = ?

First, note that the function only applies to positive inputs, so you know your answer will be positive.

To solve for a, you go through a similar substitution process as presented earlier.
Step 1: Plug a into the function and arrive at f(a) = (a + 3)2 – 5
Step 2: Set the second expression equal to 59: (a + 3)2 – 5 = 59
Step 3: Solve for a: (a + 3)2 = 64 → (a + 3) = 8 → a = 5

Certain function questions will give you two functions and ask you to use the output of one function as the input for the other function. Look at the following example:
For all numbers, x, f(x) is defined by f(x) = x2 + 12, and g(x) is defined by . What is the value of f(g4)?
This question wants you to use the output of g(4) as your input for the function f.

When working with a compound function, always go inside-out:
Step 1: Solve for g(4): .
Step 2: Substitute the output of g(4) into the function f: f(3) = 32 + 12 = 21.

Symbolism
A different type of function question will define a formula using symbols rather than the notation covered in the previous section. Such questions are often a source of intimidation for students, but the process for these questions is almost identical to the process for normal function questions.

If the operation * is defined for all numbers, a, by a* = 2a3, and (b + 3)* = 54, then b = ?
Step 1: Understand the formula: you should substitute whatever is in front of the multiplication sign for a in the formula 2a3.
Step 2: Substitute (b + 3) for a: 2(b + 3)3 = 54.
Step 3: Solve for b:

Let’s do another symbolism question with two symbols:
If the operation # is defined for all numbers, a and b, by , then 60 # (25 # 9) = ?
Step 1: Understand the formula: multiply the terms before and after the symbol and then take their square root.
Step 2: Since 60 # (25 # 9) is a compound function, you should first find the value of (25 # 9) and then input that value into the original function: .
Step 3: Solve for .

Sequences
The last type of formula question you will see are sequences.

A sequence is any group of numbers whose order is determined by a rule.

Consecutive integers are an example of a sequence: for example, the rule for the set of consecutive integers 3, 4, 5, 6, 7 is that any given term in the set is one more than the term before it.

Mathematically, the previous sequence would be defined in the following way:
an = an−1 + 1, where n > 1

The subnotation refers to the position of a term in the sequence. So a1 is the first term of the sequence, a2 is the second term, and so on. The rule tells you that the nth term of the sequence equals the value of previous term plus 1.

Why does it specify “where n > 1”? Because this rule cannot apply to the first term, since there is no number preceding that term. Thus starting with the second term, any term will have a value one greater than the term before it.

Generally, sequence questions will ask you to determine one of three things:
1. The rule for a series of numbers
2. The value of a specific term in the sequence
3. The sum or difference of two or more terms in the sequence

Example 1: The sequence a1, a2, a3, . . . , an, . . . is such that an = 2an−1 for all n > 1. If a2 = 7, what is a5?

SOLUTION: The rule is that any given term is double the term before it. a5 is 3 places after a2, so

Since a2 = 7, you know that a5 = 8 × 7 = 56.
 

Example 2: The sequence a1, a2, a3, . . . , an, . . . is such that for all n > 2. If a3 = 12 and a4 = 5, what is a2?
SOLUTION: Substitute the value of a4 for an, the value of a3 for an−1 and a2 for an−2:

Example 3: In the sequence S, s1 = 4, s2 = 11, and s3 = 18. Which of the following could be the definition of the sequence?
sn = sn−1 + 4
sn = 2sn−1 + 3
sn = sn−1 + 7
sn = 2sn−1 – 4
sn = sn−1 – 7

SOLUTION: Substitute the three values into each of the choices, and determine which choice maintains the values for all three terms:
A: 11 = 4 + 4? No → Eliminate Choice A.
B: 11 = 24 + 3? Yes
18 = 211 + 3? No
→ Eliminate Choice B.
C: 11 = 4 + 7? Yes
18 = 11 + 7? Yes
→ Keep Choice C.
D: 11 = 24 – 4? No → Eliminate Choice D.
E: 11 = 4 – 7 No → Eliminate Choice E.