Question

Determine whether the function f: (0, infinity) to (0, infinity) defined by f (x) = x^ is an injection or a surjection or a bijection​

Answer 1

Answer:

Here, f:[0,∞)→[0,∞) i.e, domain is [0,∞) and codomain is [0,∞)

For one-one 

f(x)=1+xx

f′(x)=(1+x)21>0,∀x∈[0,∞)

∴f(x) is increasing in this domain. Thus f(x) is one-one in its domain

For onto (we find range)

f(x)=1+xx i.e. y=1+xx

⇒y+yx=x

⇒x=1−yy

⇒x=1−yy≥0 as x≥0

∴0≤y=1 and y<1

∴ Range = Codomain

∴f(x) is one-one but not onto

Step-by-step explanation:

I hope it's help you