9. Introduction to Functions

A function is a rule that takes an input and gives exactly one output — like a machine: put a number in, get a number out.

Function notation \(f(x)\) is read 'f of x'. It does not mean multiplication; \(x\) is the input. The rule \(f(x) = 2x + 1\) says 'double the input and add 1'.

Evaluating means substituting a value for \(x\):

• \(f(3) = 2(3) + 1 = 7\).
• \(f(0) = 2(0) + 1 = 1\).
• \(f(-2) = 2(-2) + 1 = -3\).

You can think of \(y\) and \(f(x)\) as the same thing: \(y = 2x + 1\) and \(f(x) = 2x + 1\) graph the identical line. The input \(x\) is the horizontal value; the output \(f(x)\) is the height.

Real-world meaning. If \(C(m) = 30 + 0.10m\) gives a phone bill for \(m\) minutes, then \(C(100) = 30 + 0.10(100) = \$40\) is the cost of 100 minutes.

⚠️ Common mistake: reading \(f(3)\) as \(f \times 3\). It means 'evaluate the function when the input is 3'.

💡 Tip: whatever is inside the parentheses replaces every \(x\) in the rule.