Question

Find a bijection f:(a,b) --->(0,1)

Answer 1

Answer on Question#53954 – Math – Real Analysis

Question. Find a bijection $f \colon (a, b) \to (0, 1)$.

Solution. We shall find $f$ in the next form: $f = kx + c$. Using $\begin{cases} f(a) = 0 \\ f(b) = 1 \end{cases}$ we have:

$$\begin{cases} k a + c = 0 \\ k b + c = 1 \end{cases} \Rightarrow \begin{cases} k a + c = 0 \\ k (b - a) = 1 \end{cases} \Rightarrow \begin{cases} k = \dfrac{1}{b - a} \\ c = -\dfrac{a}{b - a} \end{cases} \Rightarrow f(x) = \dfrac{x - a}{b - a}.$$

Answer. $f(x) = \dfrac{x - a}{b - a}$.

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