If f (f(x)) = x + 1 for all x ∈ R and if f(0)

Question:

If $f(f(x))=x+1$ for all $x \in R$ and if $f(0)=\frac{1}{2}$, then $f(1)=$ ____________.

Solution:

Given: $f(f(x))=x+1$ for all $x \in R$ and $f(0)=\frac{1}{2}$

$f(f(x))=x+1$

$\Rightarrow f(f(0))=0+1$

$\Rightarrow f\left(\frac{1}{2}\right)=1 \quad\left(\because f(0)=\frac{1}{2}\right) \quad \ldots(1)$

Now,

$f\left(f\left(\frac{1}{2}\right)\right)=\frac{1}{2}+1$

$\Rightarrow f(1)=\frac{1+2}{2} \quad\left(\because f\left(\frac{1}{2}\right)=1\right)$

$\Rightarrow f(1)=\frac{3}{2}$

Hence, $f(1)=\underline{\frac{3}{2}}$.

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