Evaluating and Comparing Functions

Domain and Range: The Function's Territory

Imagine you're the manager of a movie theater. You need to know two crucial things: who can buy tickets (your potential customers) and what seats are actually available (what you can offer them). Functions work the same way—they have inputs they can accept and outputs they can produce.

When we look at a function's graph, we're seeing its complete story. The domain tells us all the x-values (inputs) the function can handle, while the range shows us all the y-values (outputs) the function can produce.

Reading Domain from a Graph

To find the domain, imagine sliding a vertical line across the entire graph from left to right. Every x-coordinate where this line touches the function is part of the domain.

Consider a parabola that opens upward with its vertex at (-2, 1) and extends infinitely in both horizontal directions. The domain would be "all real numbers" because you can draw a vertical line at any x-value and it will intersect the parabola.

Reading Range from a Graph

For range, imagine sliding a horizontal line up and down across the graph. Every y-coordinate where this line touches the function is part of the range.

Using that same parabola with vertex at (-2, 1): since it opens upward, the lowest point is y = 1 at the vertex, and it extends upward forever. The range is "y ≥ 1" or all real numbers greater than or equal to 1.

Gaps and Holes Matter

Not all graphs are continuous. If you see a gap, hole, or the graph suddenly stops, those missing pieces aren't included in the domain or range. Think of it like a broken bridge—you can't travel where there's no road.

Key Takeaway: Just like our movie theater manager needs to know their customer base and available seats, understanding domain and range gives you complete control over what a function can do. Master this, and you'll never be surprised by what inputs a function accepts or what outputs it delivers.