1. warm up
f(x) = x g(x) = 4 - x
(f о g)(x)=
(g о f)(x)=

3. Horizontal translations change both the rules and the intervals of piecewise functions. Vertical translations change only the rules.
Caution

4. Example 1: Transforming Piecewise Functions2
– x if x < 0
Given f(x) = write the
rule g(x), a vertical translation up 3. 1 2
x if x ≥ 0

5. Example 1: Transforming Piecewise Functions
2x if x < 2
Given f(x) = write the
rule g(x), a horizontal translation left 2.
x if x ≥ 2

8. Intercepts x intercept replace y with zero and solve y intercept
replace x with zero and solve

9. Find the X and Y intercepts for the piecewise function before and after the transformation1 2
– x if x < 0
Given f(x) = write the
rule g(x), a vertical translation up 3. 1 2
x if x ≥ 0

10. Find the X and Y intercepts for the piecewise function before and after the transformation
2x if x < 2
Given f(x) = write the
rule g(x), a horizontal translation left 2.
x if x ≥ 2

12. In mathematics, an _________ __________ is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x.
Example:
f(x) = x+4 [(1,5),(2,6),(3,7),(4,8)]
f-1(x) = x-4 [(5,1),(6,2),(7,3),(8,4)]

13. To write the rule for the inverse of a function, you can exchange x and y and solve the equation for y. Because the value of x and y are switched, the domain of the function will be the range of its inverse and vice versa.

14. Find the inverse of . Determine whether it is a function, and state its domain and range.

15. Find the inverse of f(x) = x3 – 2
Find the inverse of f(x) = x3 – 2. Determine whether it is a function, and state its domain and range.

16. Definition of Inverse Function
Let f and g be two functions such that
f(g(x)) = x for every x in the domain of g
g(f(x)) = x for every x in the domain of f

17. Find the inverse function of f(x) = 8x
Then verify that both f(f-1(x)) and f-1(f(x)) are equal to the identify function

18. Which of these functions is the inverse of f(x) = 7x+4?
g(x) = (x-4)/7 h(x) = (x-7)/4

20. one-to-one function
a function f is ______ ___ ______ if each x value corresponds to exactly one y value.
When both a relation and its inverses are functions, the relation is a one-to-one function.
**They pass the horizontal and vertical line test. **

21. Determine by composition whether each pair of functions are inverses.
f(x) = 3x – 1 and g(x) = x + 1

22. Determine by composition whether each pair of functions are inverses.3 2
f(x) = x + 6 and g(x) = x – 9

23. Process of finding an inverse
Use the horizontal line test to decide whether f has an inverse function
in the equation for f(x) replace f(x) by y.
interchange x and y, and solve for y
verify that f(g(x))=x and g(f(x)) = x

24. f(x) = 3x+1

25. f(x) = x3+1

26. f(x) = x3 -1