What is Domain and Range?

What is Domain and Range?

In mathematics, a function is a rule that assigns a unique output to each input. For example, the function f(x) = 2x + 3 takes any number x and multiplies it by 2, then adds 3. A function can be represented in different ways, such as a formula, a table, a graph, or a word problem.

The domain of a function is the set of all possible inputs that the function can accept. The range of a function is the set of all possible outputs that the function can produce. To find the domain and range of a function, we need to consider both the mathematical and the real-world constraints on the inputs and outputs.

How to Find the Domain of a Function

The domain of a function depends on the type of function and the context of the problem. Here are some general steps to find the domain of a function:

• Identify the type of function: linear, quadratic, polynomial, rational, radical, trigonometric, exponential, logarithmic, etc.
• Check for any restrictions on the inputs based on the type of function. For example:
  • For rational functions, the denominator cannot be zero.
  • For radical functions, the radicand (the expression under the root) cannot be negative.
  • For logarithmic functions, the argument (the expression inside the log) must be positive.
• Check for any restrictions on the inputs based on the real-world context of the problem. For example:
  • For a function that models the height of a projectile, the input (time) cannot be negative.
  • For a function that models the area of a rectangle, the input (length or width) must be positive.
• Write the domain in interval notation or set-builder notation. Interval notation uses brackets or parentheses to indicate whether an endpoint is included or excluded from the domain. Set-builder notation uses curly braces and symbols to describe the domain.

Example 1

Find the domain of the function f(x) = 1/(x - 2).

Solution:

This is a rational function. The denominator cannot be zero, so we need to exclude x = 2 from the domain. The domain is all real numbers except 2. We can write it in interval notation as:

(-∞, 2) ∪ (2, ∞)

or in set-builder notation as:

{x | x ≠ 2}

Example 2

Find the domain of the function g(x) = √(x + 4).

Solution:

This is a radical function. The radicand cannot be negative, so we need to solve for x + 4 ≥ 0. The solution is x ≥ -4. The domain is all real numbers greater than or equal to -4. We can write it in interval notation as:

[-4, ∞)

or in set-builder notation as:

{x | x ≥ -4}

How to Find the Range of a Function

The range of a function depends on the type of function and how it behaves for different inputs. Here are some general steps to find the range of a function:

• Identify the type of function: linear, quadratic, polynomial, rational, radical, trigonometric, exponential, logarithmic, etc.
• Check for any patterns or trends in the outputs based on the type of function. For example:
  • For linear functions, the range is usually all real numbers unless there is a horizontal line.
  • For quadratic functions, the range depends on whether the parabola opens up or down and what is its vertex (minimum or maximum point).
  • For polynomial functions, the range depends on whether the degree (highest power) is even or odd and what are its end behaviors (how it behaves as x approaches ±∞).
  • For rational functions, the range depends on whether there are any horizontal or slant asymptotes (lines that the graph approaches but never touches).
  • For radical functions, the range depends on whether there are any restrictions on the outputs based on the index (even or odd) of the root.
  • For trigonometric functions, such as sine and cosine, the range is usually [-1, 1] unless there are any transformations (shifts or stretches).
  • For exponential functions, such as y = b^x where b > 0 and b ≠ 1 ,the range is usually (0, ∞) unless there are any transformations.
  • For logarithmic functions, such as y = log_b(x) where b > 0 and b ≠ 1 ,the range is usually all real numbers unless there are any transformations.
• Check for any restrictions on the outputs based on the real-world context of the problem. For example:
  • For a function that models the height of a projectile, the output (height) cannot be negative.
  • For a function that models the population growth, the output (population) must be positive and integer.
• Write the range in interval notation or set-builder notation.

Example 3

Find the range of the function f(x) = 2x + 3.

Solution:

This is a linear function. The graph is a line with a slope of 2 and a y-intercept of 3. The line extends infinitely in both directions, so there is no limit on the outputs. The range is all real numbers. We can write it in interval notation as:

(-∞, ∞)

or in set-builder notation as:

{y | y is a real number}

Example 4

Find the range of the function g(x) = -x^2 + 4.

Tweet

Read More..