Functions and Transformations

Rates of change

Average rate of change

\frac{f (b) - f (a)}{b - a}

on the interval from a to b

Difference quotient

\frac{f (x + h) - f (x)}{h}

h ≠ 0

Sum and difference

(f \pm g) (x) = f (x) \pm g (x)

Product

(f \cdot g) (x) = f (x) g (x)

Quotient

(\frac{f}{g}) (x) = \frac{f (x)}{g (x)}

g(x) ≠ 0

Composition

(f \circ g) (x) = f (g (x))

Inverse functions

f (f^{- 1} (x)) = x and f^{- 1} (f (x)) = x

Even function

f (- x) = f (x)

symmetric about the y-axis

Odd function

f (- x) = - f (x)

symmetric about the origin

Horizontal shift

x - h

right h if h > 0, left |h| if h < 0

Vertical shift

+ k

up k if k > 0, down |k| if k < 0

Vertical stretch or shrink

a

by |a|; reflects across the x-axis if a < 0

Horizontal stretch or shrink

b

by 1/|b|; reflects across the y-axis if b < 0

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