Q20 of 26 Page 1

Let f: n arrown be a function defined as f (x) = 9x^2 + 6x-5 Show that f:n arrows where S is the range of f, is invertible. Find the inverse of f and hence find f^-1 (43) and f^-1 (163)

Given: a function f(x) defined as


To prove: f: N → S is invertible where S is the range of f


To find: the inverse of f and f-1 (43) and f-1 (163)


Let f(x) = y = 9x2 + 6x – 5


⇒ y = 9x2 + 6x + 1 – 1 – 5


⇒ y = (3x + 1)2 – 6


{∵ a2 + 2ab + b2 = (a + b)2}


⇒ (3x + 1)2 = y + 6






Squaring both sides:




This shows that f(x) is invertible if the range of the function is N, i.e. {1, 2, 3, 4, …………………………}


The inverse of the function f(x),


y = f(x)


⇒ f-1(y) = x


So,




Put y = 43 and 163:




Values of f-1(43) and f-1(163) are 2 and 4 respectively


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