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Sir how to show the bijection of the question D1 part(4) And how to find the inverse

Sir how to show the bijection of the question D1 part(4)
And how to find the inverse

Question Image
Grade:12

1 Answers

Aditya Gupta
2080 Points
2 years ago
te above ques is really easy.
observe that f’(x)= 2x+1 which is greater than zero for all x in domain of f. so this means that in [0,3] f is an increasing function, which proves that it is an injective function.
now, f(0)= 1 and f(3)= 9+3+1= 13. also, f is a continuous function, so it attains all the values between 1 and 13. hence, we have proved that it is surjective too.
since the function is both injective and surjective, it is a bijection by definition. hence proved
now to find the inverse, simply interchange x and y.
x=y^2+y+1
or y^2+y+1 – x=0
or y= [-1+-root(1 – 4(1-x))]/2
but since y is positive, we drop the negative sign
so, y=f-1(x)=[–1+root(4x – 3))]/2, domain of f-1 being [1,13] and range being [0,3]

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