Question

Which statement best describes the function below?

f(x)= x^3-x^2-9x+9



A. It is not a function.
B. It is a many-to-one function.
C. It fails the vertical line test.
D. It is a one-to-one function.

Answer 1

B. It is a many-to-one function.

Explanation:

It is a function because for every x values there is only one correspondent y value. Therefore it pass the vertical line test. So, option A and C are not correct. In the figure attached, a plot of the function can be seen. It is notice that different x values gives the same y value. In consequence, it is a many-to-one function. One-to-one functions are those in which every y value correspond to exactly one x value.

Answer 2

It is a many-to-one function ⇒ answer B

Explanation:

* To solve this problem lets revise some important notes

- We use the vertical line to check the graph is function or not

# If the vertical line cuts the graph in any part of it in only 1 point

then the graph represents a function

# If the vertical line cuts the graph in any part of it in more than 1

point then the graph doesn't represent a function

- We use the horizontal line to check the graph is one-to-one function

or many-to-one function

# If the horizontal line cuts the graph in any part of it in only 1 point

then the graph represents one-to-one function

# If the horizontal line cuts the graph in any part of it in more than 1

point then the graph represents many-to-one function

* Now lets use these notes to solve the problem

- Look to the attached graph

∵ The vertical lines x = -2 and x = 2 intersect the graph of f(x) in

only one point

∵ Any vertical line will cut the graph of f(x) in only one point

∴ f(x) is a function

- So answers A and C are not true, because it succeeds the vertical

line test and it is a function

∵ The horizontal lines y = -4, y = 4, and y = 11 intersect the graph of f(x)

in more the one point one point

∴ f(x) is many-to-one function

- So answer D is not true, because f(x) is many-to-one function

∴ Answer B is true because f(x) is many-to-one function