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प्रश्न
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
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उत्तर
Injectivity of f :
Let x and y be two elements of domain (R), such that
f(x) = f(y)
⇒ 4x + 3 = 4y + 3
⇒ 4x = 4y
⇒ x = y
So, f is one-one.
Surjectivity of f :
Let y be in the co-domain (R), such that f(x) = y.
⇒ 4x + 3 = y
⇒ 4x = y -3
⇒ `x = (y-3)/4 in ` R (domain)
⇒ f is onto.
So, f is a bijection and, hence, is invertible.
Finding f -1
Let f-1 (x) = y ....... (1)
⇒ x = f (y)
⇒ x = 4y + 3
⇒ x − 3 = 4y
⇒ `y = (x -3)/4`
So, `f^-1 (x) = (x-3)/4` [ from (1) ]
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