range

(noun)

the set of values (points) which a function can obtain

Related Terms

Examples of range in the following topics:

  • Probability

    • For a continuous random variable $X$, the probability of $X$ to be in a range $[a,b]$ is given as:
    • As shown below, the probability to have $x$ in the range $[\mu - \sigma, \mu + \sigma]$ can be calculated from the integral

  • Inverse Functions

    • Instead of considering the inverses for individual inputs and outputs, one can think of the function as sending the whole set of inputs—the domain—to a set of outputs—the range.
    • Let $f$ be a function whose domain is the set $X$ and whose range is the set $Y$.
    • Then $f$ is invertible if there exists a function $g$ with domain $Y$ and range $X$, with the following property:
    • Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range $Y$, in which case the inverse relation is the inverse function.

  • Average Value of a Function

    • Note that the average is equal to the area under the curve, $S$, divided by the range:
    • The average of a function $f(x)$ that has area $S$ over the range $[a,b]$ is $\frac{S}{b-a}$.

  • Area of a Surface of Revolution

    • If the curve is described by the parametric functions $x(t)$, $y(t)$, with $t$ ranging over some interval $[a,b]$ and the axis of revolution the $y$-axis, then the area $A_y$ is given by the integral:
    • The spherical surface with a radius $r$ is generated by the curve $x(t) =r \sin(t)$, $y(t) = r \cos(t)$, when $t$ ranges over $[0,\pi]$.

  • Calculus of Vector-Valued Functions

    • A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors.
    • The dimension of the domain is not defined by the dimension of the range.

  • Inverse Functions

    • For this rule to be applicable, for a function whose domain is the set $X$ and whose range is the set $Y$, each element $y \in Y$ must correspond to no more than one $x \in X$; a function $f$ with this property is called one-to-one, or information-preserving, or an injection.
    • If the domain is the real numbers, then each element in the range $Y$ would correspond to two different elements in the domain $X$ ($\pm x$), and therefore $f$ would not be invertible.

  • Vectors in the Plane

    • This plane may be described parametrically as the set of all points of the form$\mathbf R = \mathbf {R}_0 + s \mathbf{V} + t \mathbf{W}$, where $s$ and $t$ range over all real numbers, $\mathbf{V}$ and $\mathbf{W}$ are given linearly independent vectors defining the plane, and $\mathbf{R_0}$ is the vector representing the position of an arbitrary (but fixed) point on the plane.

  • The Natural Logarithmic Function: Differentiation and Integration

    • The Taylor polynomials for $\ln(1 + x)$ only provide accurate approximations in the range $-1 < x \leq 1$.

  • Inverse Trigonometric Functions: Differentiation and Integration

    • Note that some of these functions are not valid for a range of $x$ which would end up making the function undefined.

  • The Derivative as a Function

    • Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions.