Understanding Function Notation

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 :\(g(7)= 2(7)^2 - 6\)
The function outputs the value 92: \(g(7)=92\)

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