Unit 6: Functions

Interval Notation

Set Notation

{x<0}. {-4<x<0}

How to identify the domain of a function algebraically?

1. Terms under a radical can’t be less than 0. Set up a number line and test integers.

2. Denominators can’t equal 0.

3. Radical in denominator: set expression under the radical greater than 0.

Domain and Range: (y = x)

D: all real numbers

R: all real numbers

Domain and Range: (y = x²)

D: all real numbers

R: [Vertex, positive/negative infinity)

Domain and Range: (y = radical x)

D: [0, positive infinity), R: [0, positive infinity)

Domain and Range: (y = lxl)

D: all real numbers, R: [0, positive infinity)

Domain and Range: (y = x³)

D: all real numbers, R: all real numbers

Domain and Range: (y = 1/x)

D: all real numbers except 0, R: all real numbers except 0

When the value is added outside the function

Shifts up

When the value is subtracted outside the function

Shifts down

When the value is added inside the function

Shifts left

When the value is subtracted inside the function

Shifts right

When multiplying by a constant on the outside

Stretches the graph, multiply the y’s

When multiplying by a negative on the outside

Flips vertically over horizontal axis

When multiplying by a negative on the inside

flips horizontally over vertical axis

y = k(x-h)

Shift right h units

y = k(x+h)

Shift left h units

y = k(x)+h

Shift up h units

y = k(x)-h

Shift down h units

y = ak(x)

Multiply the y’s by a

y = -k(x)

Reflect over the x-axis

y = k(-x)

Reflect over the y-axis

Average range of change

Slope formula

When determining when the graph is increasing or decreasing

Look at the x coordinates

Absolute Maximum

Highest point in graph

Absolute Minimum

Lowest point in graph

Local maximum

A high point in the middle of the graph

Local minimum

A low point in the middle of the graph

(f + g)(x)

f(x) + g(x)

(f - g)(x)

f(x) - g(x)

(f * g)(x)

f(x) * g(x)

(f/g)(x)

f(x) / g(x)

Composition function

f * g(x) = f(g(x))

g * f(x) = g(f(x))

Inverse functions

Functions that reverse each other's effects, such that if f(g(x)) = x and g(f(x)) = x.

f^-1(x) = f(x)

Calculating the inverse of f(x)

1. Switch x and y

2. Solve for y

3. If a function, replace y with f^-1(x)

One-to-one Function

Every input has one output and every output has one input. Must pass the vertical line and horizontal line test.

When do we use a domain restriction when finding the inverse of f(x)?

When the range of the original function doesn’t match the domain of the inverse we find.