1. 1.4 Building Functions from Functions

2. If f(x) and g(x) be functions with intersecting domains, then:
Sum:
Difference:
Product:
Quotient:
are all functions

3. For f(x) = 3x + 5 and g(x) = 2x – 1, find the following with each domain.
1. f(x) + g(x)
2. f(x) – g(x)
3. (fg)(x)
4. f(x)
g(x)

4. Defining new functions algebraically
Let and
Find formulas for the following functions and give the domain of each:
f + g
f – g fg gg

5. Composition of Functions
The composition f of g, is defined:

6. For f(x) = 3x + 5 and g(x) = 2x – 1, find f(g(x))

7. Composing functions Let and .
Find and , and verify then functions are not the same.

8. Finding the domain of a composition
Let and
Find the domain of the composite functions: 1) 2)

9. You Try!
If and find:

10. Decomposing Functions
For each function h, find functions f and g such that h(x)=f(g(x))

11. Decomposing Functions
For each function h, find functions f and g such that h(x)=f(g(x))

12. You Try!
For function h, find functions f and g such that h(x)=f(g(x))

13. Horizontal Line Test
The inverse of a relation is a function if and only if the original relation passes the horizontal line test. Meaning, each horizontal line intersects the graph of the original relation in at most one point.

14. One-to-One Functions
A function for which every element of the range of the function corresponds to exactly one element of the domain Passes both the vertical and horizontal line test.

15. Inverse Functions
If f is a one-to-one function with domain D and range R, then the inverse function of f, denoted f-1, is the function with domain R and range D defined by f-1(b)=a if and only if f(a)=b

16. Inverse Functions Only 1-1 functions possess inverse functions
These functions have range elements that correspond to only one domain element each, so there's no danger that their inverses will not be functions.
The horizontal line test is a quick way to determine whether a graph is that of a one-to-one function.

17. Find an equation for f-1(x) if f(x)=x/(x+1)

18. If , find

19. You Try! Find the inverse of:1) 2)

20. Finding an inverse function graphically
To find the inverse, find the reflection of the given graph across the line y=x.

21. The Inverse Composition Rule
A function f is 1-1 with inverse function g if and only if:
f(g(x))=x for every x in the domain of g
and
g(f(x))=x for every x in the domain of f

22. Verifying inverse functions
Show algebraically that and are inverse functions.

23. You Try!
Confirm algebraically that f and g are inverses by showing that f(g(x))=x and g(f(x))=x