Question

A function f is defined from R → R as f(x) = ax + b, such that f(1) = 1 and f(2) = 3. Find function f(x). Hence, check whether function f(x) is one-one and onto or not.

If a function f : X → Y defined as f(x) = y is one-one and onto, then we can define a unique function g : Y → X such that g(y) = x, where x ∈ X and y = f(x), y ∈ Y. Function g is called the inverse of function f. The domain of sine function is R and function sine : R → R is neither one-one nor onto. The following graph shows the sine function. Let sine function be defined from set A to [– 1, 1] such that inverse of sine function exists, i.e., sin⁻¹ x is defined from [– 1, 1] to A. On the basis of the above information, answer the following questions :

Let f : R+ → [– 5, ∞) be defined as f(x) = 9x² + 6x – 5, where R+ is the set of all non-negative real numbers. Then, f is :

A relation R is defined on N × N (where N is the set of natural numbers) as : (a, b) R (c, d) ⇔ a – c = b – d. Show that R is an equivalence relation.

A function f : ℝ → ℝ defined as f(x) = x² - 4x + 5 is :