Show that the function f : R →R , defined as f (x) = x2 , is

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 Multiple Choice QuestionsShort Answer Type

381.

Let * be a binary operation on the set Q of rational numbers given as a * b = (2a – b)2, a, b ∈ Q. Find 3 * 5 and 5 * 3. Is 3 * 5 = 5 * 3 ?

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382. Consider f : R+ → [4, ⊂) given by f(x) = x+ 4. Show that f is invertible with the inverse f–1 of f given by straight f to the power of negative 1 end exponent left parenthesis straight y right parenthesis equals square root of straight y minus 4 end root where R+ is the set of all non-negative real numbers.
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383.

Let f : R → R be defined as f (x) = 3 x. Choose the correct answer. (A) f is one-one onto   (B) f is many-one onto  (C) f is one-one but not onto (D) f is neither onc-one nor onto.

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384.

how that the function f : R* → R* defined by

straight f left parenthesis straight x right parenthesis equals 1 over straight x is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*?

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385. Show that the function f : R →R , defined as f (x) = x2 , is neither one-to-one nor onto.


Here f(x) = x2 Df = R Now 1,–1 ∈ R Also f (1)= 1, f (–1) = 1 Now 1 ≠ –1 but f (1) = f (–1)
∴ f is not one-to-one.
Again, the element – 2 in the co-domain of R is not image of any element x in the domain R.
∴ f is not onto.

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386.

Show that the modulus function f : R → R, given by f (x) = | x |, is neither one-one nor onto, where |x| is x, if x is positive or 0 and | x | is —x, if. x negative.

 

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387. Find f(A), if f(x) = x2–5x –14 and straight A equals space open square brackets table row 3 cell negative 5 end cell row cell negative 4 end cell 2 end table close square brackets
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 Multiple Choice QuestionsLong Answer Type

388.

Let A= R × R and * be a binary operation on A defined by

(a, b) * (c, d) = (a+c, b+d)

Show that * is commutative and associative. Find the identity element for *

on A. Also find the inverse of every element (a, b) ∈ A.

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389.

Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary operation on A defined by (a, b) * (c, d) = (ac, b+ad) for (a, b), (c, d) element of A. Then find
(i) The identify element of * in A.
(ii) Invertible elements of A, and write the inverse of elements (5, 3) and open parentheses 1 half comma space 4 close parentheses.

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390. Let space straight f colon straight W rightwards arrow straight W space be space defined space as
straight f left parenthesis straight n right parenthesis space equals space open curly brackets table row cell straight n minus 1 comma space space if space straight n space is space odd end cell row cell straight n plus 1 comma space if space straight n space is space even end cell end table close curly brackets
Show that f is invertible and find the inverse of f. Here, W is the set of all whole numbers. 
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