range

(noun)

The set of values (points) which a function can obtain.

Related Terms

  • Range
  • function
  • domain

(noun)

The set of values the function takes on as output.

Related Terms

  • Range
  • function
  • domain

Examples of range in the following topics:

  • Visualizing Domain and Range

    • Example 1:  Determine the domain and range of each graph pictured below:
    • The range for the graph f(x)=−112x3f(x)=-\frac{1}{12}x^3f(x)=−​12​​1​​x​3​​, is R\mathbb{R}R.
    • Example 2: Determine the domain and range of each graph pictured below:
    • The range of the blue graph is all real numbers, R\mathbb{R}R.
    • Use the graph of a function to determine its domain and range
  • Introduction to Domain and Range

    • The domain of a function is the set of all possible input values that produce some output value range
    • A function is the relation that takes the inputs of the domain and output the values in the range.
    • As you can see in the illustration, each value of the domain has a green arrow to exactly one value of the range.  
    • The oval on the left is the domain of the function fff, and the oval on the right is the range.  
    • The green arrows show how each member of the domain is mapped to a particular value of the range.
  • Inverse Trigonometric Functions

    • Note that the domain of the inverse function is the range of the original function, and vice versa.
    • Note the domain and range of each function.
    • $\displaystyle{y = \cos^{-1}x \quad \text{has domain} \quad \left[-1, 1\right] \quad \text{and range} \quad \left[0, \pi\right]}$
    • To find the domain and range of inverse trigonometric functions, we switch the domain and range of the original functions.
    • Describe the characteristics of the graphs of the inverse trigonometric functions, noting their domain and range restrictions
  • One-to-One Functions

    • In other words, every element of the function's range corresponds to exactly one element of its domain.
    • Another way to determine if the function is one-to-one is to make a table of values and check to see if every element of the range corresponds to exactly one element of the domain.  
    • The ordered pairs (−2,4)(-2,4)(−2,4) and (2,4)(2,4)(2,4) do not pass the definition of one-to-one because the element 444 of the range corresponds to to −2-2−2 and 222.
  • Piecewise Functions

    • Example 2: Graph the function and determine its domain and range:
    • The range begins at the lowest yyy-value, y=0y=0y=0 and is continuous through positive infinity.  
    • Therefore the range of the piecewise function is also the set of all real numbers greater than or equal to 000, or all non-negative values: y≥0y \geq 0y≥0.
  • Inequalities with Absolute Value

    • If both are solved for xxx, we will see the full range of possible values of xxx.
    • We now have two ranges of solutions to the absolute value inequality; x>4x > 4x>4 and x<−4x < -4x<−4.
  • Graphs of Logarithmic Functions

    • The range of the function is all real numbers.
    • The range of the square root function is all non-negative real numbers, whereas the range of the logarithmic function is all real numbers.
  • Solving Problems with Logarithmic Graphs

    • Firstly, doing so allows one to plot a very large range of data without losing the shape of the graph.
    • Secondly, it allows one to interpolate at any point on the plot, regardless of the range of the graph.
    • A key point about using logarithmic graphs to solve problems is that they expand scales to the point at which large ranges of data make more sense.
  • Inverses of Composite Functions

    • If fff is an invertible function with domain XXX and range YYY, then
  • Introduction to Variables

    • Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation.
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