1. Precalculus 1.7
INVERSE FUNCTIONS

2. DO THIS NOW!
You have a function described by the equation: f(x) = x + 4
The domain of the function is: {0, 2, 5, 10}
YOUR TASK: write the set of ordered pairs that would represent this function
**use the equation to find f(x) for each number in the domain
Graph the points in the function.

3. Functions as Sets of Ordered Pairs
Recall that besides describing functions with equations or graphs, we can do so by listing the ordered pairs that make up the function

4. Inverse function Notation: f-1 is the inverse of f
Definition: A function’s inverse brings output values back to their input values.
f(x)=x+4 6 2
INPUT
OUTPUT
f-1(x)=x-4

5. Inverse Functions: 3 representations
GRAPH
ALGEBRA
ORDERED PAIRS

6. Inverse Functions: Ordered Pairs
Original function, f: {(0, 4) (2, 6) (5, 9) (10, 14)}
Inverse function, f-1: {(4, 0) (6, 2) (9, 5) (14, 10)}
What is f(2)?
What is f-1(6)?
To find the inverse function, represented by ordered pairs, simply flip each ordered pair
If f contains (x, y), then f-1 contains (y, x).

7. Inverse Functions: Algebra
The equation of the inverse function should “undo” the equation of the original function.
Ex: If f(x) = x + 4, then f-1(x) = x – 4
Ex: If g(x) = 4x, then g-1(x) = …?

8. Precise Definition
Let f and g be two functions such that 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. Then g is the inverse of the function f. g = f-1 The domain of f is the range of f-1. The range of f is the domain of f-1.

9. A logical point:
If f is the inverse of g, then g is the inverse of f. Furthermore, g and f are inverses of each other.

10. Example
Which of these is the inverse of:

11. Inverse Function: GRAPHS
Graph these functions and their inverses on the same graph.
f(x) = x and g(x) = x – 4
f(x) = 4x and g(x) = x/4
f(x) = .5(x+3) and g(x) = 2x – 3
f(x) = x2 and
{(0,1) (2,5) (3,6) (4,8)} and {(1,0) (5,2) (6,3) (8,4)}

12. Inverse Graphs The graph of a f and f-1 are related in a special way.
If (x,y) is on f’s graph, (y,x) must be on f-1’s graph.
Therefore, the graph of f-1 is a reflection of the graph of f across the line y = x.

13. Practice What is the inverse of: f(x) = 2x + 4

14. Which functions have inverses?
What is this function’s inverse? (0, 1) (2, 4) (3, 4)

15. Functions without inverses
If multiple input values have the same output value, then the function has no inverse
This is because given a repeated output, there would be no way to tell what the original input was
f: (0,1) (2,4) (3,4)
f-1: (1,0) (4,2) (4,3) This is NOT a function! Therefore we have no surefire way to undo f.

16. Functions without inverses: a list
f(x) = x2 (What was x if f(x)=4?)
f(x) = xn where n is even
f(x) = |x|

17. Functions without inverses: graph test
If a function, f, has an inverse, f-1, then the inverse is also a function. Therefore f-1 must pass the vertical line test. In order for f-1 to pass the vertical line test, f must pass the HORIZONTAL LINE TEST.
f-1 f

18. Will these functions have inverses?1) 2) 3) 4)
If a function both increases and decreases,
can it have an inverse?

19. Finding the inverse of a function algebraically
Use the horizontal line test to decide whether f has an inverse.
In the equation, replace f(x) with “y.”
Switch “x” and “y.”
Solve for y.
Replace y with “f-1(x).”
Check your work! 1. 2. 3. 4. 5.

20. Practice: find the inverses of these functions1) 2) 3) 4) 5) 6)