In further maths, the range of a function refers to all the possible output values (y-values) that the function can produce. This is according to the definition: the range of a function refers to all the possible values y could be. The formula to find the range of a function is y = f(x).
Here's a breakdown of the concept:
Understanding Range
The range is the set of all possible dependent variable values (usually 'y') that result from using the function with all possible independent variable values (usually 'x'). In other words, if you input all possible 'x' values into the function, the range is the set of all the 'y' values you get out.
Finding the Range
Finding the range of a function depends on the function itself. Here are some common approaches:
- Consider the function's behavior: Analyze the function's equation and determine any limitations on the possible output values.
- Graph the function: The range can be easily determined from the graph as the set of all y-values the graph covers.
- Use the domain: The domain is the set of all possible input values (x-values). Sometimes, knowing the domain helps restrict the possible output values.
- Algebraic manipulation: You might be able to rearrange the equation to solve for x in terms of y. This can help you identify any restrictions on the possible values of y.
Examples
Example 1: Linear Function
Consider the function f(x) = 2x + 1. Since x can be any real number, 2x + 1 can also be any real number. Therefore, the range is all real numbers, often written as (−∞, ∞).
Example 2: Quadratic Function
Consider the function f(x) = x2. Since the square of any real number is non-negative, the range is all non-negative real numbers, often written as [0, ∞).
Example 3: Rational Function
Consider the function f(x) = 1/x. x can be any real number except 0. f(x) can take on any value except 0. Therefore, the range is all real numbers except 0, often written as (−∞, 0) ∪ (0, ∞).
Range vs. Codomain
It is important to differentiate between the range and the codomain of a function. The codomain is the set within which the output of the function must fall. In many cases, the codomain and range can be the same, but not always. The range is more specific; it's the actual set of outputs produced by the function, while the codomain is a broader declaration of where those outputs are expected to be.
