1. Inverses of Functions Part 2
Lesson 2.9

2. Reminder from yesterday
Add to the cue column in your notes:
When graphing the 𝑓 −1 𝑥 , choose points 𝑓(𝑥) on either side of the vertex

3. Vertical Lines
Choose 4 points on the vertical line below. Write the coordinates for each.
What IS a vertical line?
A line composed of all the points _________________.
with the same x value

4. Vertical Line Rule
Why do we use a vertical line to determine if a graph is a function?
If a vertical line passes through 2 points on a graph, both those points have the same ____ value but 2 different ___ values. Therefore, the ______ has _______________ output. This means the graph _____ a function. In a function, each x value has ________ y value.
only one
What kind of line did we draw here?
What is occurring at the circled points?
What do we know about the x values of these two points?
What does that mean about whether the graph is a function? Stress RIGHT is RIGHT. x y
input
more than one
is not
only one

5. Practice Draw the line 𝑥=3 through the graph.
Circle the points where it crosses the graph. Write the coordinates of those points.
What does this tell you about this graph?
Why does the vertical line rule determine if a graph is a function?

6. Is the Inverse of the Function a Function?
Draw the inverse of the function below.
Use the vertical line rule to determine if the inverse of f is a function.

7. The horizontal line rule
Horizontal line in the original becomes a vertical line in the inverse:
𝑦=−3
𝑥=−3

8. The horizontal line rule (cont’d)
If a horizontal line drawn through ______________ crosses more than one point, _______________ is not a function.
At these two points, the graph of the original has one ____________ with multiple ______________. So, its inverse will have one ________ with multiple _________
the original graph
its inverse
𝑦=−3
output
inputs
input
outputs
𝑥=−3

9. Horizontal v. Vertical Line Test
Use vertical line test when we have the graph of the inverse
Use horizontal line test when we only have the graph of the original function, and not the inverse

10. Practice
1. Determine whether the following graphs are functions.
WITHOUT drawing the inverse graph, determine whether the inverse of the relation is a function.
Graph 3 - function
𝑓(𝑥)
not a function
function
function
𝒇 −𝟏 (𝒙)
function
not a function
function

11. Defend your answer
Graph the inverse.
Use the vertical line rule.

12. Independent Practice
Complete Problem Set independently.