Pre-Calc Unit 1

function

equation where each x corresponds to only one y, though each y can correspond to multiple x values

one-to-one function

function where each y matches to only one x value

f^-1(x)

f(x) flipped over y = x axis

Steps for inverting a function

1. switch x and y

2. solve for y

3. isolate y

PEMDAS when inverting a function

IS REVERSED

Horizontal line test:

draw horizontal line through function graph to determine if it is a one-to-one or many-to-one

Many-to-one function can’t be

inverted, because then it will go from failing the horizontal line test (which a function can do) to failing the vertical line test (which a function can’t do)

Vertical-line-test

draw a vertical line through the graph of a function. it should not intersect any one x twice- if it does, it is not a function

find if g is the inverse of f

insert one into the other (the inserted one takes the place of x of the one it is inserted into), then solve for x. If x can be isolated, it is inverse

hole

discontinuity where f(x) is removable by factoring, like if it is in the numerator AND denominator

Ex: x + 2 / (x + 2)(x - 4)

x = -2 is a hole

asymptote

discontinuity that can’t be factored out of equation

discontinuity

part of graph where x can’t equal a certain value

numerator discontinuity

x ≠ 0

sqrt discontinuity

under sqrt can’t = negative

jump

piecewise functions

domain set notation

(n, n)

(n, n)U(n, n)

() in set notation

not including

[ ] in set notation

including

f o g =

f of g of x

f(g(x))

g o f =

g of f of x

g(f(x))

add g(x) and f(x)

f(x) + g(x)

(f + g)x

subtract f(x) and g(x)

f(x) - g(x)

(f - g)x

multiply f(x) and g(x)

f(x) * g(x)

(f * g)x

divide f(x) and g(x)

f(x) / g(x)

(f / g)x

** g(x) != 0

Operating with functions graphically

plug in x of first function, then act as if the y of that one is the x of the next

Ex: f(g(x)) = f(x) → g(x) where g’s input equals f’s output

Even

f(x) = f(-x)

Even graph

Symmetrical across y axis

Odd

f(-x) = -f(x)

Odd graph

rotated 180 degrees around origin

When -f(x), do you make the denominator negative?

NO, because -f(x) = -1(f(x)) = -1/1*(f(x))

Translation

moves f(x) up/down, right/left

Scale

makes f(x) larger or smaller

Reflection

Flips f(x) across an axis

Vertical shrink

0 < a < 1

Vertical stretch

a > 1

Horizontal shrink

a > 1

Horizontal stretch

0 < a < 1

Vertical stretch/shrink compared to parentheses

a(x)

Horizontal stretch/shrink compared to parentheses

(ax)

First step with quadratics

FACTOR

• x should have no coefficient

• x should be positive

Vertical reflection

-f(x)

across x-axis

Horizontal reflection

f(-x)

across y-axis

Vertical shift

(x) +-a

up/down

Horizontal shift

(x +-a)

right/left