Question

Show that the function f : N → N, given by f(x) = 2x, is one-one but not onto.​

Answer 1

Answer:

Solution The function f is one-one,

for f(x1) = f(x2) ⇒ 2x1= 2x2 ⇒ x1 = x2

Further,f is not onto, as for 1 ∈ N, there does not exist any x in N such that f(x) = 2x = 1.

Answer 2

Answer:

Since we have given that

f: N to N is defined by

f(x)=2x-1f(x)=2x−1

For one- one, let x₁, x₂ ε N

So, Let f(x₁) = f(x₂)

\begin{lgathered}2x_1-1=2x_2-1\\\\2x_1=2x_2\\\\x_1=x_2\end{lgathered}

2x

1

−1=2x

2

−1

2x

1

=2x

2

x

1

=x

2

So, it is one-one.

Now, let y = 2x-1

So, we get that

\begin{lgathered}y+1=2x\\\\\dfrac{y+1}{2}=x\end{lgathered}

y+1=2x

2

y+1

=x

But if we let y = 2

then, \dfrac{2+1}{2}=\dfrac{3}{2}=1.5

2

2+1

=

2

3

=1.5

And we know that 1.5 does not belong to natural numbers.

Hence, it is one-one but not onto.

# learn more:

F:N to N defined by f(m)=m^2+m+3 is one one function

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