Functions and Function Notation

A function is a special relationship where each x-value (input) corresponds to exactly one y-value (output). Think of it as a machine that processes inputs from the domain to produce unique outputs in the range. You can quickly identify functions using the vertical line test - if any vertical line crosses the graph more than once, it's not a function.

Function notation like f(x) = 3x²-5x+2 gives us a powerful way to express relationships. When you see something like f(2), it's asking "what's the y-value when x = 2?" To find it, you simply substitute 2 for x in the function and calculate. For example, calculating f(-2) means plugging -2 into the function: 3(-2)² - 5(-2) + 2 = 12 + 10 + 2 = 24.

A one-to-one function passes both the vertical line test AND the horizontal line test, meaning each output comes from exactly one input. These functions are particularly useful because they can be reversed - each y-value leads back to exactly one x-value.

💡 When solving problems like "find f(x) = 3," you're actually being asked "what x-value(s) give a y-value of 3?" This is different from finding f(2), which asks for the output when the input is 2.