We already know that a function is a special type of relation where for each x-value there is one and only one y-value. With a one-to-one function, we can also say that for each y-value there is one and only one x-value. So for each x, there is one y and for each y, there is one x. A horizontal line of the form y = k, can be used to determine if a function is one-to-one. For each point on a horizontal line, the y-value is the same, so if any horizontal line impacts the graph in more than one location, this tells us that the given y-value is associated with more than one x-value and is not a one-to-one function.

Test Objectives
• Demonstrate an understanding of the concept of a one-to-one function
• Demonstrate the ability to sketch the graph of a function
• Demonstrate the ability to determine if a function is one-to-one using the horizontal line test
Horizontal Line Test Practice Test:

#1:

Instructions: determine if the function is one-to-one.

$$a)\hspace{.2em}f(x)=\frac{1}{3}x - 1$$

#2:

Instructions: determine if the function is one-to-one.

$$a)\hspace{.2em}f(x)=(x + 1)^3 + 1$$

#3:

Instructions: determine if the function is one-to-one.

$$a)\hspace{.2em}f(x)=|x - 1| - 1$$

#4:

Instructions: determine if the function is one-to-one.

$$a)\hspace{.2em}f(x)=(x - 2)^2$$

#5:

Instructions: determine if the function is one-to-one.

$$a)\hspace{.2em}f(x)=\sqrt{x + 1}+ 3$$

Written Solutions:

#1:

Solutions:

a) This function is one-to-one.

#2:

Solutions:

a) This function is one-to-one.

#3:

Solutions:

a) This function is not one-to-one.

#4:

Solutions:

a) This function is not one-to-one.

#5:

Solutions:

a) This function is one-to-one.