Miss Young
Pre-Calculus
Fall 2014
Function Operations and Composition of Functions
1) (𝑓 + 𝑔)(𝑥) = _____________________
1) (𝑓 ∘ 𝑔)(𝑥) = _______________________
2) (𝑓 − 𝑔)(𝑥) = _____________________
2) (𝑔 ∘ 𝑓)(𝑥) = _______________________
3) (𝑓𝑔)(𝑥) = _____________________
𝑓
4) (𝑔) (𝑥) = ____________________
Example: Find all four operations using the following functions:
𝑓(𝑥) = 2𝑥 − 3 and 𝑔(𝑥) = 𝑥 2 − 1.
1)
2)
3)
4) *(Watch for the domain)
Example: Find both compositions using the following functions:
𝑓(𝑥) = 𝑥 + 2 and 𝑔(𝑥) = 4 − 𝑥 2
1)
2)
Miss Young
Pre-Calculus
Fall 2014
𝑓(𝑥) = 𝑥 2 − 9 and √9 − 𝑥 2 (**watch the domains)
1)
2)
Inverse Functions
Notation:
When f(x) is composed with its inverse, you obtain the identity function which is ___________.
To obtain the inverse from a given set of points, switch the _____ and _____ values.
(_____________ values switch with the _____________ values)
When finding the inverse from a given function, switch the ____ and ____ and solve for _____.
When finding the inverse from a graph, the point (a, b) on the graph of the function becomes
(____, _____) on the graph of the inverse function. The function is reflected across the line of
symmetry (y = ______)
Example: Find the inverse function of 𝒇(𝒙) = 𝟒𝒙
Determine which of the following is the inverse of (𝒙) =
𝑔(𝑥) =
𝑥−2
5
5
and ℎ(𝑥) = 𝑥 + 2
𝟓
𝒙−𝟐
:
Miss Young
Pre-Calculus
Fall 2014
Sketch the graphs of the inverse functions 𝒇(𝒙) = 𝒙𝟐 + 𝟏 ; 𝒙 ≥ 𝟎 and 𝒇−𝟏 (𝒙) = √𝒙 − 𝟏
One – to – One Functions
________________________________ - a function f has an inverse function if and only if no
_______________ line intersects the graph of f at more than one point.
__________________ functions – if each value of the dependent variable corresponds to
exactly one value of the independent variable. A function f has an inverse function if and only if
it is so. (No y - value is matched with more than one x – value)
Example:
Make an x, y table for 𝑓(𝑥) = 𝑥 2 and determine if it will be one-to-one and whether or not it
will have an inverse function.