Consider f: R+ → [−5, ∞) given by f(x) = 9x^{2} + 6x − 5. Show that f is one one,onto hence invertible

f: R+ → [−5, ∞) given by f(x) = 9x^{2} + 6x − 5

to show that function is one-one. we will show it by the contradiction.

let us take function is not one-one. therefore there exist two or more numbers for which images are same.

we will take two different numbers

let

since are positive. therefore it cannot be zero.

therefore

therefore it contradicts our assumption. hence the function is one-one.

A function f:X→Y is onto if for every y∈Y there exist a preimage in X.

let y= 9x^{2} + 6x − 5

hence f(x)=y therefore f(x) is onto.

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