precalc study sheet

circle equation

(x-h)^2 + (y-k)^2 = r^2

what is the difference between parallel and perpendicular lines

parallel lines have the same slope, perpendicular lines have the opposite reciprocal slope

what is an inverse function

An inverse function is a function whose x values have been switched for its y values, and vice versa.

what does a one-to-one function mean

no repeating inputs or outputs

vertical line test

If any vertical line passes through no more than one point of the graph of a relation, then the relation is a function.

horizontal line test

If any horizontal line only crosses at a graph at most once, it has an inverse function and is one-to-one

rules for inverse functions (3)

must be one-to-one

range and domain of the og is domain and range of the inverse

graphs are reflections over y = x

how to find an inverse function algebraically

1. replace f(x) with y

2. solve the equation for x in terms of y

3. swap every x with y and every y with x

4. replace y with f-1(x) for the final inverse function

f(x) = x

domain and range

identity function

straight up line

domain = real numbers

range = real numbers

f(x) = absolute value x

domain and range

triangular

absolute value function

domain = (- infinity, infinity)

range = [o, infinity)

f(x) = x^2

domain and range

parabola

quadratic function

domain = (- infinity, infinity)

range = [o, infinity)

f(x) = x^3

cubic function

squiggly

domain = (-infinity, infinity)

range = [o, infinity)

f(x) = √x (domain/range)

radical function

one sided curve upwards

[o, infinity) x2

f(x) = 1/x Domain and Range

rational function

two curves that never touch

(- infinity, infinity) [2, infinity)

vertical shift

y = (x) + k

horizontal shift

y = (x+h)

vertical dilation

y = a(x)

multiply y values by a

horizontal dilation

y = bx

divide x values by b

rules of horizontal dilation

shrinks/compresses when abs value b is more than one

stretches in x direction when it is between 0 and 1

reflection about x axis

-f(x)

(x,y) -> (x,-y)

reflection about y axis

f(-x)

(x,y) -> (-x,y)

even function

graph is symmetrical across the y-axis; f(x) = f(-x)

opposite inputs have the same output

odd function

origin symmetry, f(-x)=-f(x)

opposite inputs have opposite outputs