Change in Tandem: Understanding Functions

A function maps input values to output values with one critical rule: each input value must correspond to exactly one output. Think of it as a machine where you put in an x-value and get exactly one y-value out.

Functions have two key components: the independent variable $x-values or inputs$ and the dependent variable $y-values or outputs$. When analyzing functions graphically, remember that a function is positive when its graph is above the x-axis and negative when below.

Understanding how functions change is crucial. A function is increasing when output values rise as input values increase. Conversely, it's decreasing when output values fall as input values increase. Additionally, functions can be concave up (rate of change increasing) or concave down (rate of change decreasing).

Pro Tip: When identifying intervals where a function is increasing or decreasing, always express your answer using the input values $x-values$, not the outputs!

For example, if a piecewise function is shown on the interval [0,9], and we need to find where it's increasing, we'd look for sections where the graph rises from left to right—in this case (0,3) and (7,9).