# Math 4 Unit 2: Functions and Their Graphs

#### Vocab

Domain – the span of all the x-values of a function

Range – the span of all the y-values of a function

x-intercept(s) – the point(s) where a function touches the x-axis

y-intercept – the point where a function touches the y-axis

Horizontal asymptote(s) – the imaginary line(s) where the domain of a rational function approaches on both sides but never touches.

• There are three rules that define where the asymptote will be and how it will look like:
• If the degree of the numerator is less than the degree of the denominator, the asymptote will, by default, be y = 0.
• The asymptote will be higher or lower if there is a k value in the function.
• If the degree of the numerator is equal to the degree of the denominator, the asymptote will, by default, be at y = (degree of the numerator/degree of the denominator)
• The asymptote will be higher or lower if there is a k value in the function.
• If the degree of the numerator is greater than the degree of the denominator, there will not be a horizontal asymptote.

Vertical asymptote – the imaginary line(s) where the range of a rational function approaches above and below but never touches.

Local minimum – the point where a positive even polynomial’s vertex is, where the function does not go any lower.

Local maximum – the point where a negative even polynomial’s vertex is, where the function does not go any higher.

End behavior – the direction, attributes, and/or limit of a function as it goes left or right.

#### Application

The height of a passenger on an escalator is modeled as

where t is the time in seconds after the passenger has boarded the escalator on the second floor and h is the height in feet above the first floor.

This means it takes the passenger 16 seconds to reach the first floor (0 feet).

This means the passenger is 12 feet above the first floor when he/she first gets on the escalator. Likewise, this also means that the second floor is 12 feet above the first floor.

#### Inverse Functions

Find the inverse of the following function:

#### Composition of Functions

Find f(g(x)) and its domain if:

and:

#### Decomposition of Functions

Find what h(x) decomposes into if:

and: