range

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 $f(x)=-\frac{1}{12}x^3$, is $\mathbb{R}$.
  • Example 2: Determine the domain and range of each graph pictured below:
  • The range of the blue graph is all real numbers, $\mathbb{R}$.
  • 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 $f$, 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 $(-2,4)$ and $(2,4)$ do not pass the definition of one-to-one because the element $4$ of the range corresponds to to $-2$ and $2$.

• Piecewise Functions

  • Example 2: Graph the function and determine its domain and range:
  • The range begins at the lowest $y$-value, $y=0$ 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 $0$, or all non-negative values: $y \geq 0$.

• Inequalities with Absolute Value

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

• 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 $f$ is an invertible function with domain $X$ and range $Y$, 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.