Real functions | Part-2 | find range of a function

Finding the Range of a Function

The range of a function refers to the set of all possible output values (dependent variables) that the function can produce for its defined inputs. Understanding how to determine the range is essential when analyzing real-valued functions.

Understanding the Range of a Function

In simple terms, the range includes all values that a function can return when values from its domain are used as inputs.

Notation: If f: A → B is a function, the range is a subset of B.

Steps to Find the Range of a Function

1. Determine the domain: Find all input values where the function is valid and defined.
2. Express the function algebraically: Use the equation of the function to explore its behavior.
3. Apply constraints: Consider mathematical restrictions such as square roots, division, or logarithms.
4. Solve for output: Rearrange the function (if possible) to express x in terms of y, then determine valid values for y.
5. Use graphing (optional): A graph of the function can help visualize the possible output values.

Illustrations for Real-Valued Functions

1. Polynomial Function

Example: f(x) = x²

Domain: ℝ

Range: [0, ∞) since squaring any real number results in a value that is zero or positive.

2. Rational Function

Example: f(x) = 1 / (x² + 1)

Domain: ℝ

Range: (0, 1] since x² + 1 ≥ 1 and as x increases, f(x) approaches 0 but never reaches it.

3. Square Root Function

Example: f(x) = √(x – 2)

Domain: x ≥ 2

Range: [0, ∞) because the smallest output is 0 (when x = 2), and values increase as x increases.

4. Logarithmic Function

Example: f(x) = log(x)

Domain: x > 0

Range: ℝ, because the logarithmic function can output any real number depending on the value of x.

Key Points

• The range includes every possible output that the function yields for inputs in its domain.
• Identifying constraints in the function helps in finding the range.
• Graphical methods can support analytical techniques to find the range.
• The type of function—whether it's polynomial, rational, square root, or another—affects how its range behaves.