# Properties

Mon 01 January 2018

## Properties

- Associative property: a + b + c = (a + b) + c = a + (b + c) and abc = (ab)c = a(bc)
- Commutative property: a + b = b + a and ab = ba
- Distributive property: a(b + c) = ab + ac

Addition commutative, which means that a + b = b + a. It is also associative, which means that if you have a long summation like a + b + c you can compute it in any order (a + b) + c or a +

## Formulas

\(\frac{1}{x^{2}}\) is the same as \(x^{-2}\)

## Common functions and their inverses

$$f(x) ⇔ f^{-1}({x})$$

$$2^x ⇔ \log_2(x)$$

Following can be read as scale x by 3 times and add five and the inverse would be subtract 5 and scale x down by a factor of 3.

$$3x + 5 ⇔ \frac{1}{3}(x - 5)$$

Returns `e^x`

, the inverse function of `\log(x)`

.

$$a^x ⇔ log_a (x)$$

$$exp(x) ≡ e x ⇔ ln(x) ≡ log_e(x)$$

$$sin(x) ⇔ sin−1 (x) ≡ arcsin(x)$$

$$cos(x) ⇔ cos−1 (x) ≡ arccos(x)$$

The function-inverse relationship is reflexive—

###### Completing the square

Any quadratic expression \(Ax^2 + Bx + C\) can be rewritten in the form \(A(x − h)^2 + k\) for some constants h and k. Geometrically h shifts the graph right and k up.