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Question

The function f:R+(1,e) defined by f(x)=X2+eX2+1 is

A
One-one but not onto
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B
Onto but not one-one
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C
Both one-one and onto
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D
Neither one-one nor onto
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Solution

The correct option is C Both one-one and onto
f(x)=x2+ex2+1f(x)=2x(x2+1)2x(x2+e)(x2+1)2=2x3+2x2x32ex(x2+1)2=2x2xe(x2+1)2=2x(1e)(x2+1)2<0f(x)<0,f(x) is decreasing
Hence f is one-one function.
x0,f(x)e
x,f(x)e
Hence range = (1, e) = co-domain



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