Master the concept of dependent and independent variables in functions for CBSE Class 11 Applied Mathematics with solved examples and practice sets.
A function is a rule that assigns each input value from a set to exactly one output value. The input variable, typically denoted by x, is called the independent variable because it can be chosen freely from the domain. The output variable, denoted by y or f(x), is called the dependent variable because its value is determined by the rule applied to x.
Step 1: Identify the input variable x as the independent variable.
Step 2: Identify the output f(x) as the dependent variable.
Step 3: For x = 2, f(2) = 3(2) + 5 = 11. Here, 11 is the dependent value.
Answer: Independent: x, Dependent: f(x).
Functional notation f: A → B indicates that f maps elements from set A to set B. For every x ∈ A, there exists a unique y ∈ B such that y = f(x). This relationship ensures that for a single input, there is no ambiguity in the output.
Step 1: Substitute x = 3 into the expression.
Step 2: Calculate f(3) = (3)² - 4.
Step 3: Simplify to get 9 - 4 = 5.
Answer: f(3) = 5.
The set of all possible values for the independent variable x is the domain, while the set of all resulting values for the dependent variable y is the range. Understanding this distinction is crucial for graphing functions and analyzing their behavior.
Step 1: Calculate f(1) = 2(1) = 2.
Step 2: Calculate f(2) = 2(2) = 4.
Step 3: Calculate f(3) = 2(3) = 6.
Answer: Range = {2, 4, 6}.