In the study of relationships between two variables, the value of one variable (known as the dependent variable) often relies on the value of another (the independent variable). Functions provide a way to understand how one variable, often called the dependent variable, is related to another variable, known as the independent variable.

**The Relationship Between Variables**

Consider the relationship between sales and revenue. As sales increase, typically, so does revenue. This is a common example of a functional relationship, where revenue (R) can be considered a function of sales (X):

R=4X

In this scenario, R is the dependent variable whose value is determined by the value of X, the independent variable.

**Identifying Functions**

A crucial aspect of functions is understanding when an equation truly represents a function. The test is simple: if each value of the dependent variable corresponds to exactly one value of the independent variable, then the equation is a function.

**Examples of Functions**

- Linear functions, such as
*y*=2*x*+3 - Quadratic functions, like
*y*=*x*−4^{2} - Exponential functions, exemplified by
*y*=x^{2}

**Non-Function Equations**

- Circles, such as x
^{2}+y^{2}=9, because a single x-value can correspond to two y-values. - Vertical lines, like
*x*=4, as they do not pass the vertical line test.

**Graphical Representation of Functions**

The standard practice for representing functions in a graph is to place the independent variable on the horizontal axis and the dependent variable on the vertical axis. This convention facilitates the use of graphical analysis, including the implementation of the line test to make sure whether a relationship is a function.To determine if a relationship in a graph is a function, we can employ the Vertical Line Test. This test stipulates that a graph represents a function if, and only if, any vertical line drawn through the graph intersects it no more than once. Conversely, the Horizontal Line Test is used to verify functions by ensuring that any horizontal line cuts through the graph of the relationship at most once.

**Domain and Range: The Breadth and Depth of Functions**

The domain of a function encapsulates all permissible values of the independent variable for which the function is defined.

**Determining the Domain**

To find the domain:

- For linear functions, such as
*x*+3, the domain is all real numbers, (−∞,∞)(−∞,∞). - For polynomial and quadractic functions, like 6
*x*^{3}+4*x*^{2}+7*x*−7, the domain also spans all real numbers. - For rational functions, such as 1 / (x -4) the domain excludes values that make the denominator zero, hence
*x*≠ 4. - For radical functions, the radicand must be greater than or equal to zero. For instance, for the square root of x-4 , the domain is [4,∞)

**The Range of Functions**

The range refers to all possible values that the dependent variable can take:

- For linear functions, the range is going all real numbers, (−∞,∞)(−∞,∞).
- For the function y = x
^{2 }with the leading power of 2 , it is going to have a limited infinity.The range for this equation is [0, ∞).Similarly, for x^{2}+4 will have the range of [4,∞). ^{For the quadratic function like x2 +2x+3, we use the formula to determine the x-coordinate of the vertex. In the standard form of a quadratic function, ax2 +bx+c, the coefficients are a=1, b=2, and c=3. Applying the formula, we find that the x-coordinate of the vertex is −1. Substituting x=−1 back into the original equation, we find the y-coordinate: f(−1)=(−1)2+2(−1)+3=2. Therefore, the vertex of the function is at the point (−1,2). For a quadratic function where a > 0, the range is from the y-coordinate of the vertex to infinity. Hence, the range of this function is [2,∞)}.- The range of a cubic polynomial, or any polynomial with a leading term that has an odd exponent, spans all real numbers.y For eg: y = x
^{3}+ 5x^{2 }+2x+1. The range for these equations is going to be (-∞ , ∞).

**Interval Notation: Communicating Domains and Ranges**

Interval notation is a way of writing subsets of the real numbers that express intervals. Closed intervals, denoted by square brackets [], include their endpoints, while open intervals, denoted by parentheses (), do not. This notation is used to concisely describe the domain and range of functions.

**Finite Intervals**: The closed interval [2,5] includes the endpoints 2 and 5, while the open interval (2,5)excludes them.**Infinite Intervals**: An interval like (−∞,7) includes all numbers less than 7, but not 7 itself.