By the end of this lesson, you should:

1. Understand function notation and its use cases

2. Be able to distinguish between relations and functions

3. Solve problems involving function notation

Functions are used throughout mathematics and function notation was developed to make it easier to work with them.

Let \(f\) denote a given function and let \(x\) represent the input value. \(f(x)\) denotes the output value when \(x\) is the input. Basically, \(f(x) = y\), and so, \(f(x)\) is often used in place of \(y\) when describing the equation of a function. It is important to note here that \(f(x)\) is only used to represent functions and not all equations. For example, the equation \(y=x\) can also be written as \(f(x) = x\), but the equation \(y = \pm\sqrt{1-x^2}\) cannot be written using \(f(x)\) because it is not a function. In fact, \(y = \pm\sqrt{1-x^2}\) is the equation of a circle of radius 1, centered at the origin, and therefore does not pass the vertical line test, indicating that it is not a function.

When evaluating a function, input the given \(x\)-value into the equation of the function and solve to produce the output. Evaluating the function, \(f(x)\), at \(x=a\) will produce the output of \(f(a)\).

Evaluate the function \(f(x)=2x^2-7x+3\) at \(x=6\).

\(f(6) = 2 \times\ (6)^2 - 7 \times\ (6) + 3\)

\(f(6) = 64 - 42 + 3\)

\(f(6) = 25\)

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