# Understanding Function Notation

Functions are introduced as math studies get more abstract. The function takes an **input value** and produces an **output value** according to a **rule**. However, functions are just a new way of looking at familiar topics.

## Function Notation

While doing math, you might see something like this:

\[f(x) = 5x - 4\]The left side *f(x)* looks like a monomial term of “*f* times *x*”. Instead, this is **function notation** and reads “**f of x** equals 5 times *x* minus 4”. One clue here is that the monomial “*f* times *x*” would typically be written as *fx* (without parentheses) or (*fx*) within parentheses.

## What “f” Means

The letter **f** is **not a variable** but rather a **name for the function**. Once a function is defined as above, later in a problem the function might simply be referred to by its letter name, *f*. A new function will be given a new letter, such as *g(x)*—read “*g* of *x*”.

## The Purpose of Functions

A function is a different way of referring to an equation. The equation \(y = 5x - 4\) is immediately recognized as having a straight line graph. The graph of the function \(f(x) = 5x - 4\) is the same line. Instead of being asked “for the equation \(y = 5x - 4\) find the value of \(y\) when \(x = 2\)” the equivalent question in “function language” is “for the function \(f(x) = 5x -4\), find \(f(2)\).”

## Evaluating Functions

The above function is **evaluated the same way as an equation**, by substituting 2 for \(x\): \(5(2) - 4 = 6\). Instead of reporting the answer as “\(y = 6\)”, the answer is “\(f(2) = 6\)”. The **input value** is \(2\), and the **output value** is \(6\).

Similarly, other problems given in function notation can be translated to more familiar terms.

Other examples:

**Example 1:** \(f(x) = 19 - 5x\)

*Input* the value \(4\) for \(x\): \(f(4) = 19 - 5(4)\)

The function *outputs* the value \(-1\): \(f(4) = -1\)

**Example 2:** \(g(x) = 2x^2 -6\)

*Input* the value 7 for *x* :\(g(7)= 2(7)^2 - 6\)

\(g(7)=98-6\)

The function *outputs* the value 92: \(g(7)=92\)

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