Square Root Functions and Their Graphs

Square root functions like f(x) = √$x-3$ behave differently from regular functions because they can't handle negative numbers under the radical. When evaluating this function, we get real answers only when whatever's inside the square root is zero or positive.

Let's explore what happens when we plug in different values: f(28) = √(28-3) = √25 = 5, and f(3) = √(3-3) = √0 = 0. But when we try f(2) = √(2-3) = √(-1), we hit a problem—you can't take the square root of a negative number in the real number system. This restriction creates the domain of the function, which is x ≥ 3 or [3, ∞).

Finding the domain of square root functions follows a simple pattern: whatever is inside the square root must be greater than or equal to zero. For example, with f(x) = √$4x-24$, we solve 4x-24 ≥ 0, which gives us x ≥ 6. Similarly, for f(x) = √$10-2x$, we get 10-2x ≥ 0, meaning x ≤ 5.

Try This! When graphing square root functions, the point where the expression inside the square root equals zero becomes your starting point $often the x-intercept$. From there, the graph extends only in the direction allowed by the domain.