1. New Functions from Old
Section 1.3

2. Stretches
Multiplying a function f(x) by a constant c stretches the graph vertically if c>1 or shrinks the graph if c<1
A negative c flips the graph over the x axis in addition to stretching or shrinking

3. Vertical Shifts
Replacing y with (y-k) shifts the graph up by k or down if k is negative.
If y = x2 then y-4 = x2 =>y = x2 + 4
and the graph is shifted up 4

4. Horizontal Shift
Replacing x with x - h moves the graph to the right by h, if h is positive and to the left if h is negative.
y = x2 then y = (x-3)2 is shifted to the right by 3

5. Composite Functions f•g(x) = f(g(x))
f of g of x: Substitute x into g and get an answer; then substitute the answer into f
Domain of f•g is all x in the domain of g that have results that are in the domain of f

6. odd and even functions f is an even function, if f(-x) = f(x)
f is an odd function, if f(-x) = -f(x)
f is neither if f is neither odd or even

7. Notes on Inverse Functions
Section 1.3

8. What is an Inverse Function
If a function f takes x and gives y then the inverse function takes y and gives x.
Notation:

9. Domain and Range of Inverses
If a function has an inverse:
The domain f-1 is the range of f
The range of f-1 is the domain of f
f and f-1 “undo” each other
If a function has an inverse it is said to be invertible

10. How to tell if a function has an Inverse
Horizontal line test:
If you can pass a horizontal line up and down the graph of the function and it only hits the graph in one point everywhere then it has an inverse.
The graph is “Strictly increasing” or “Strictly decreasing”
Unique value test:
Each value of x in the domain of f get paired with one and only one value

11. How to find an inverse: Case 1: T-Chart
If you are given a T-chart of values of f
Test to see if f is one-to-one to make sure f-1 exists
Create f-1 by reversing the x and y values

12. How to find and inverse Case 2: Graph
The graph of f-1 is a reflection of f in the line y=x.
Sketch y=x on the graph
Find points on f and reverse x and y
Plot the new points

13. How to find and inverse Case 3: formula
If you are given f(x)= expression
Replace f(x) with y
Swap the x’s and y throughout the formula
Solve for y
Replace y with f-1 (x)