Function Range

Definition

The range of a function $f$ is the set of all resulting output values (dependent variable $y$) that the function can produce from its domain.

• How to read: “The range is the set of all resulting y values.”
• Meaning: The ‘output space’ containing every value the function actually reaches.

Why It Matters

Knowing the range allows us to understand the limits of a system’s output. For example, if a control system’s range is limited to a certain voltage, expecting it to drive a motor requiring higher voltage is a design failure. It defines the “envelope of possibility” for a given transformation.

Core Concepts

• Constraints by Function Nature: The outputs are limited by the mathematical form of the function.
  • Example: The range of $\sin(x)$ and $\cos(x)$ is always between $-1$ and $1$.
    • How to read: “The range of sine x and cosine x is from negative one to one.”
    • Meaning: Trig functions cannot exceed unit-circle bounds regardless of input angle.
• Determining Range:
• Example: For $f(x) = x^2 + 4$, since $x^2 \ge 0$ for all real $x$, the range is $ — algebra and precalculus foundations.

Connected Concepts

• Function Definition, Function Notation
• Function Domain
• Inverse Functions
• Function Transformations
• Modeling with Functions