**Objectives**

- Identify intervals where a function is increasing, decreasing, or constant.
- Visually discriminate between graphs of linear and nonlinear functions.
- Use the horizontal line test to determine whether a function is one-to-one.
- Explore the graphs of special functions.

**Key Terms**

- Horizontal – Straight from left to right.
- On a coordinate plane, the equation of a horizontal line is y = a, where a is a constant.

- Horizontal Line Test – A method for testing whether the inverse of a given graph is a function.
- If any horizontal line can be drawn that intersects the graph at two or more points, then the inverse of the graph is not a function.
- If no such line can be drawn, then the inverse of the graph is a function.
- A method for testing whether a given graph is a one-to-one function.

- Many-to-One Function – A function in which 2 or more x-values map to a given y-value.
- Each
*x*-value has more than one corresponding*y*-value.

- Each
- One-to-One Function – A function in which each y-value has a single x-value mapped to it.
- No two
*x*-values map to the same*y*-value.

- No two

**Notes**

- Linear and Nonlinear Functions
- A linear function is a graph that demonstrates constant rate of growth or decline
- A curve (nonlinear function) demonstrates growth or decline that is not at a constant rate

- Linear functions are graphs of straight lines, but they must NOT be vertical lines.
- Vertical lines would fail the vertical line test, obviously.

- Nonlinear functions are graphs of anything other than straight lines.
- The graph of a curve that passes the vertical line test is a nonlinear function.
- The graph of a combination of line segments that passes the vertical line test is a nonlinear function.

Nonlinear Function (curved)

- Review: What is a function?
- Function – A graph that passes the vertical line test (each x-value goes to exactly ONE y-value)
- Draw vertical lines straight up and down on a graph.
- Do any of the lines cross the graph more than once?
- If so, it’s not a function.
- If they ONLY cross once, it IS a function!

- Do any of the lines cross the graph more than once?

- Draw vertical lines straight up and down on a graph.
- Relation – Any graph (functions and non-functions)
- The following graph is a relation, but not a function (it fails the vertical line test in many places)

- One-to-One Function – Passes the vertical line test AND the horizontal line test
- One input has exactly one output.

- Many-to-One Function – Passes the vertical line test but FAILS the horizontal line test
- On a graph: If a horizontal line crosses the graph at more than one point.
- On an input-output table: If any y-value has more than one x-value listed with it.
- Many inputs can have the same output.

- Function – A graph that passes the vertical line test (each x-value goes to exactly ONE y-value)

- Constant, Increasing, or Decreasing
- A linear function has a constant rate of change; however, it may increase, decrease, or be constant over time.
- To determine if a graph is increasing, decreasing, or constant, look at the graph from left to right. Does it go up, down, or stay the same?

Increasing

Decreasing

Decreasing (starts by going down from the left to right; and, even though it goes up and down infinitely, it ends going down on the right)

Constant

- Example: Investing
- An investment is an amount of money you start with to (hopefully) make more money.
- Interest is a fee charged by a lender or borrower for the use of borrowed money.
- When you put money in a bank account, the bank pays you interest because it is borrowing your money
- Simple interest is paid only on the original investment.
- Each year, Peter’s account would earn 8% of $1000.

- Compound interest is paid on the original investment and on any interest already earned.
- Each year, Peter’s account would earn 8% of ($1000 + interest from all previous years).

- Simple interest is paid only on the original investment.
- Which is better for Peter, simple or compound interest?

Answer: Compound interest is the best option for Peter, because he will earn more money over time.