12. Functions, Exponents, Polynomials, Quadratics, and Final Review

A function is a rule where each input has exactly one output. Function notation \(f(x)\) means "the output when the input is \(x\)." It does not mean multiplication.

If \(f(x)=2x+3\), then \(f(5)=2(5)+3=13\).

Exponents mean repeated multiplication. Key rules:
\[ x^a\cdot x^b=x^{a+b},\qquad \frac{x^a}{x^b}=x^{a-b},\qquad (x^a)^b=x^{ab}. \]
So \(x^2\cdot x^5=x^7\).

Polynomials are expressions like \(x^2+7x+12\). Factoring reverses multiplication:
\[ x^2+7x+12=(x+3)(x+4). \]

Quadratic equations may be solved by factoring or by the quadratic formula. For \(x^2-9=0\), factor:
\[ (x-3)(x+3)=0, \]
so \(x=3\) or \(x=-3\).

Final review method: after every missed problem, label the miss as one of these: reading, setup, formula, algebra move, arithmetic, calculator entry, or unit check. That turns mistakes into a study map.