Welcome to our exploration of functions and relations. In mathematics, we study relationships between sets of numbers or objects. Two fundamental concepts are relations and functions. A relation is any set of ordered pairs that connects elements from one set to elements of another set. Here we see a simple relation connecting elements from set A to set B.
Now let's define what makes a function special. A function is a particular type of relation where each input has exactly one output. The key rule for functions is that every element in the domain, or input set, must map to exactly one element in the range, or output set. This one-to-one correspondence is what distinguishes a function from a general relation.
Let's compare a valid function with an invalid one to understand the difference clearly. First, we see a valid function where each input maps to exactly one output. Now watch what happens when we violate the function rule. Here, input 2 maps to both q and r. This is not a function because it violates the fundamental rule: one input must have exactly one output only.
Let's look at mathematical examples to see functions in action. Here we have three common function formulas: f of x equals 2x plus 1, g of x equals x squared, and h of x equals square root of x. Each formula assigns exactly one output value for every valid input value. On the graph, we can see the linear function f of x equals 2x plus 1 in blue, and the quadratic function g of x equals x squared in red. Notice how each x-value corresponds to exactly one y-value on each curve.
To summarize what we've learned about functions and relations: A relation is any set of ordered pairs that connects elements from two sets. A function is a special type of relation where each input has exactly one output. Functions must follow the fundamental rule that each input maps to exactly one output. Mathematical formulas like f of x equals 2x plus 1 are common examples of functions. Functions are essential tools used throughout mathematics and science to model relationships between quantities.