bijective function from [a,b] to [c,d] Im trying to think about bijective function from the closed interval [a,b] to the closed interval [c,d]. When $a,b,c,d \in \mathbb{R}$ and $a &lt; b,\;c &lt; d$.
Is there such a function?
The idea is to construct a line whose domain is $[a, b]$ and whose range is $[c, d]$

Hence, two points on the line will be
$$
p_0 = (a, c) \\
p_1 = (b, d)
$$

Since we want $a \to c$, $b \to d$, and a straight line between them.

The slope of such a line will be
$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{d - c}{b - a}$$

Using point slope form of a line,

$$
y - y_1 = m \left ( x - x_1 \right) \\
y - c = \frac{d - c}{b - a} \left (x - a\right)
$$

Is the required equation.
$f(x)=(d-c)\frac{x-a}{b-a}+c$   is the (linear) function