RD Sharma Class 11 Maths Solutions Chapter 3 - Functions

The correct method to find the domain and range depends on the type of function. The general approach taught in RD Sharma solutions is:

• Domain: Identify all possible input values (x) for which the function is defined. Check for constraints like denominators not being zero or expressions inside square roots being non-negative.
• Range: Find the set of all possible output values (y). This can be done by expressing x in terms of y and finding the permissible values for y, or by analysing the function's graph and its minimum/maximum values.

For example, for a function f(x) = 1/√(x-2), the domain is x > 2, and the range is (0, ∞).

To solve problems involving the algebra of functions (sum, difference, product, or quotient), follow these steps:

1. First, determine the domain of each individual function, let's say D₁ for f(x) and D₂ for g(x).
2. The domain of the combined function (f+g, f-g, or f*g) is the intersection of the individual domains, i.e., D₁ ∩ D₂.
3. For the quotient function f/g, the domain is D₁ ∩ D₂ but with an additional condition: you must exclude all values of x for which g(x) = 0.

Determining the domain is critical because it defines the set of all valid input values for which the function produces a real and defined output. Every subsequent calculation, including finding the range, evaluating the function, or performing algebraic operations, depends on this initial domain. Solving a problem without first establishing the correct domain can lead to mathematically undefined or incorrect results, a common pitfall in complex RD Sharma problems.

While NCERT introduces these concepts, RD Sharma solutions tackle more complex applications. Problems in RD Sharma often:

• Combine functions: For example, finding the domain/range of f(x) = [x] + |x-2| or f(x) = 1 / ([x] - 3).
• Involve inequalities: Solving inequalities like |x-1| + |x+2| > 4, which requires a detailed, case-by-case analysis based on breaking points.
• Test deeper properties: Questions may require understanding the interaction between different function types, demanding a more advanced problem-solving approach than typical textbook exercises.

The standard method detailed in RD Sharma solutions for finding the range of a linear rational function is as follows:

1. Set the function equal to y, so y = (ax+b)/(cx+d).
2. Cross-multiply to get y(cx+d) = ax+b.
3. Rearrange the equation to isolate x. You will get an expression like x = (dy-b)/(a-cy).
4. The range consists of all real values of y for which x is defined. The only value for which x is undefined is when the denominator is zero.
5. Set the denominator (a-cy) to zero and solve for y. The range is the set of all real numbers except for this value (R - {a/c}).

A very common mistake is to only calculate the intersection of the domains of f(x) and g(x). While this is the first step, students often forget the second, crucial part: you must also explicitly find and exclude all values of x for which the denominator function, g(x), is equal to zero. Detailed RD Sharma solutions always emphasise this extra step, as overlooking it results in an incomplete and incorrect domain.

Yes, all solutions for RD Sharma Class 11 Maths Chapter 3 are meticulously prepared and verified by subject matter experts to be fully compliant with the latest CBSE syllabus and guidelines for the 2025-26 academic session. They cover all the concepts, definitions, and problem types prescribed in the curriculum.