1. Bell Ringer
1. What are the ways to solve a system of equations?
2. Where do you find the “answer” when solving a system of inequalities?
3. Are systems functions? Why?

2. Functions and Inverses
Wednesday, September 6, 2017

3. Functions
A function is a relation in which each input has exactly one output. All functions pass the vertical line test.
We will deal with Polynomial Functions (includes Liner, Quadratic, Cubic, etc.), Piecewise Functions (includes Absolute Value), Radical, Rational, Exponential, and Trigonometric Functions (includes Sine and Cosine).

4. Function or not?

5. Inverse Functions The basics:
If it’s a table or a set of ordered pairs, simply switch x and y.
If it’s a graph, the inverse is a reflection across the line y = x.
If you have the function in y = or f(x) = form, switch x and y and solve for y.

6. What do you think the inverse will be? IS The inverse a function?

7. What do you think the inverse will be? IS the inverse a function?

8. What do you think the inverse will be? Is the inverse a function?

9. What do you think the inverse will be? Is the inverse a function?

10. What do you think the inverse will be? Is the inverse a function?

11. What do you think the inverse will be? Is the inverse a function?

12. Make the inverse
Since we know visually functions and their inverses are reflections over the line y = x, we should realize that to create an inverse function, you switch x and y (from transformations in Math II. Based on that, create a table of the inverse of this function. Please use correct function notation for the inverse.

13. How do we know if a function has an inverse function?

14. Examples

15. Why???
To find an inverse function, you switch the x and y values. Since the x values are the domain of the function, the domain of your function becomes the range of the inverse and the range of the function becomes the domain of the inverse. Sometimes you must restrict the domain to insure the inverse is also a function.

17. Classwork
Is the relation a function? If so, find the inverse function; restrict the domain if necessary.
[(2, 1), (-3, 0), (1, 5)]
[(4, 5), (6, 5), (3, 5)]
[(-2, 5), (3, 7), (-2, 8)]
[(0, -1.1), (2, -3), (1.4, 2), (-3.6, 8)]
y = -5x
y = 3x – 4
y = x2

18. homework
Is the relation a function? If so, find the inverse function; restrict the domain if necessary.
[(3, 4), (4, 3), (6, 5), (5, 6)]
[(-2.5, 1), (-1, -1), (0, 1), (-1, 1)]
y = 3x
y = 7x – 6
y = 2x2 – 3

19. exit ticket
How can you visually determine if two functions are inverses?
What test lets you know if a function has an inverse function?
What might need to be restricted in a function to insure that its inverse is also a function?