AI Tutor
Imagine a vending machine where pressing button B4 always dispenses a cold seltzer — never a bag of chips. In mathematics, a function is just like that: a rule where every single valid input produces exactly one output.
In 1734, Swiss mathematician Leonhard Euler invented the symbol f(x)f(x) to write this relationship compactly, changing mathematics forever.
How do we find what comes out when we drop a specific number into the machine?

Evaluating at Numeric Inputs

To evaluate a function, replace every occurrence of the input variable with your chosen value using parentheses.
Let's evaluate f(x)=2x2−5x+3f(x) = 2x^2 - 5x + 3 at x=−3x = -3. Wrapping negative numbers in parentheses prevents sign errors when squaring.
A common mistake is writing −32=−9-3^2 = -9 instead of (−3)2=9(-3)^2 = 9. When substituting into an exponent, the negative sign is part of the base and must be squared too.
Numbers are straightforward — but what happens when you feed an entire algebraic expression into a function?

Evaluating at Variable Inputs

The input variable xx acts as an empty container or placeholder. Whatever is inside the parentheses replaces every single instance of xx in the formula.
For example, if g(x)=3x−4g(x) = 3x - 4, finding g(a+2)g(a + 2) means substituting (a+2)(a + 2) wherever you see xx.