Functions

Definition and Basic Properties

A function is a rule that assigns to each element in a set called the domain exactly one element in a set called the codomain. The set of all possible output values is called the range.

• Domain: The set of all input values (x-values) for which the function is defined.

• Range: The set of all possible output values (f(x)-values) produced by the function.

Finding Domain and Range

• For functions involving square roots, the expression under the root must be non-negative.

• For rational functions, the denominator must not be zero.

Example 1: Domain and Range

• Given:

• Domain:

• Range: Since the square root function outputs non-negative values, .

• Interval Notation: Domain: , Range:

• Given:

• Domain:

• Range: can take any real value except $0 and denominator can be any nonzero real number).

• Interval Notation: Domain: , Range:

Function Composition

The composition of two functions and is written as .

• To compute , substitute into every occurrence of in .

Example 2: Function Composition

• Given: and

• Compute:

• Solution:

• Compute:

• Solution:

Difference Quotient

The difference quotient is a fundamental concept in calculus, used to define the derivative. For a function , the difference quotient is:

• It measures the average rate of change of the function over the interval .

Example 3: Difference Quotient

• Given:

• Compute:

• Solution:

  • Difference quotient:

  • Simplified:

• Given:

• Compute:

• Solution:

  • Difference quotient:

Summary Table: Key Concepts