A function is a relation where a rule is defined to assign exactly one value to each element of a domain of values. This set of resulting values is called the range of the function.

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Note that all functions are relations, but not all relations are functions.

Unless otherwise stated, the domain, or set of "input" values is represented by the variable "x", while the set of "output" values is represented by the variable "y".

The domain variable (x) is referred to as the **independent** variable.

The range variable (y) is referred to as the **dependent** variable.

Usually function notation is used to represent a relation which is a function. Given that relation "f" is a function, it can be represented as y = f(x) (stated as "y equals f at x"). A value substituted into the function is called the argument of the function.

Unless otherwise stated, independent and dependent variable are real numbers.

One analogy used to describe a function is a "machine" that converts an input value into an output value.

Example One

**Problem**

1. Explain why **f**: 6x - 2y = 8 is a function, and write it using function notation.

2. Evaluate the function for the arguments -5, Q, 2x + 1 .

3. Graph y = f(x) and use the vertical line test to verify it is a function.

4. State the domain and range of the function **f**.

Click here for a solution.

Example Two

**Problem**

1. Explain why **g**: x2 - 2x - y - 3 = 0 is a function, and write it using function notation.

2. Evaluate the function for the arguments 4, t, 2x + 1 .

3. Graph y = g(x) and use the vertical line test to verify it is a function.

4. State the domain and range of the function **f**.

Click here for a solution.

Example Three

**Problem**

1. Explain why **S**: x = y2 is a NOT function.

2. Graph relation **S** and use the vertical line test to verify it is NOT a function.

3. State the domain and range of relation **S**.

Click here for a solution.

Example Four

**Problem**

1. Explain why **R**: x = 2y + sin(y) is a NOT function.

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2. Graph relation **R** and use the vertical line test to verify it is NOT a function.