Think of functions as a machine with inputs and outputs. If the machine is represented by the equation $x + 7$ , and the input (i.e. the value we use for $x$ ) is $2$ , then the output would be $9$ because $2 + 7 = 9$ . This is the core concept behind all function problems. People usually just get tripped up in the notation.

The function described above is represented as:

$f (x) = x + 7$

In other words, this function of $(x)$ is defined as $x + 7$ . The input is what is in the parentheses: $x$ . The machine takes the input and carries out the operation $x + 7$ to get the output. There is no $f$ variable in this equation - the $f$ simply denotes that this is a function.

The $f$ , for function, may also be written as a $g$ .

In fact, the name of the function can be any letter(s). You can always be sure you are working with a function if that name is italicized and takes something else in parentheses, like $f (x)$ , $g (y)$ , or $h (z)$ .

Try solving this simple function:

Given: $f (x) = 4 x + 2$

What is $f (3)$ ?

Do you know the answer?

(spoiler)

Answer: $14$

Plug in the $3$ for the $x$ .

$f (x) = 4 (3) + 2 = 12 + 2 = 14$

Simple enough, right?

Nested functions

Sometimes you’ll see a function nested within another function.

Given: $f (x) = x + 3$ and $g (x) = 3 x$

What is $f (g (3))$ ?

The notation may seem confusing at first, but this is the same concept as before. We just need to solve two functions: first the inner one $g (x)$ , and then we use that answer as the input to the outer $f (x)$ function.

Try solving this and then check your work below.

(spoiler)

Answer: 12

First, input $3$ into the inner $g (x) = 3 x$ function.

$g (3) = 3 * (3) = 9$

Then plug $9$ into the outer $f (x) = x + 3$ function for the final answer.

$f (9) = (9) + 3 = 12$

Multiple input functions

Sometimes a single function can have multiple inputs. For example, $f (x, y) = 2 x - y$ is a function with two inputs. The solution to $f (2, 1)$ can be found by replacing $x$ with $2$ and $y$ with $1$ .

$f (x, y) f (2, 1) f (2, 1) f (2, 1) = 2 x - y = 2 (2) - 1 = 4 - 1 = 3$

Symbolic functions

If you see a strange-looking symbol that doesn’t look like a normal math symbol, you are likely looking at a function problem in disguise. For example, you might see a question that looks like this:

What is the value of $3 ⋆$ , given that $x ⋆ = x - 2.5$ ?

Have a guess?

(spoiler)

Answer: 0.5

These problems can be translated into functions pretty easily. Even though we have a weird symbol, $x ⋆$ is really just $f (x)$ , and this turns into a basic function question.

$x ⋆ f (x) f (3) f (3) = x - 2.5 = x - 2.5 = 3 - 2.5 = 0.5$

These symbols might be hearts, boxes, stars, diamonds, or something else that looks out of place. Translate these strange symbolic functions into regular functions and they’ll be easier to approach!

Bringing it all together: question walkthrough video

Here’s a video going through one of our practice questions to demonstrate these ideas in action: