Question

Which of the statements below give 1:1 correspondence between R= {reals} and the interval (0, Pi).

A. y= COS x
B. neither answer gives 1:1 correspondence
C. x= y³
D. y= x
E. x= y²
F. y= x³
G. y= tan x
H. y= sin x
I. y= cotx
J. y=x²

Answer 1

The functions C. x = y³ and F. y = x³ establish 1:1 correspondences between the set of real numbers and the interval (0, Pi).

Step-by-step explanation:

The question is asking which function provides a 1:1 correspondence between the set of real numbers (R) and the interval (0, Pi). To establish 1:1 correspondence, each element of the first set must be paired with a unique element of the second set, and vice versa. Analysis of the options:

• A. y = COS x cannot be a correct answer because the cosine function repeats every 2π and is not one-to-one.
• B. Neither answer gives 1:1 correspondence is not a valid option.
• C. x = y³ provides a 1:1 correspondence if we limit y to (0, π), as the cube of any real number is unique.
• D. y = x is not bounded within (0, π) for all real numbers x.
• E. x = y² does not provide a 1:1 correspondence because squaring positive and negative numbers yields the same result.
• F. y = x³ also establishes a 1:1 correspondence, as cubing a real number results in a unique value.
• G. y = tan x does not establish 1:1 correspondence because the tangent function repeats every π.
• H. y = sin x cannot be a correct answer because the sine function repeats every 2π and is not one-to-one.
• I. y = cot x is like the tangent function and is not one-to-one.
• J. y = x² does not provide a 1:1 correspondence because squaring positive and negative numbers yields the same result.

By evaluating the options, the functions that provide 1:1 correspondence are C. x = y³ and F. y = x³.