y = \dfrac{x+1}{x-1}

Graph of the Linear Fractional Function $y = \dfrac{x+1}{x-1}$

$y = \dfrac{x+1}{x-1}$ is a linear fractional function, with a linear polynomial in both the numerator and the denominator. It has the general form $y = \dfrac{ax+b}{cx+d}$ with $c \neq 0$ and $ad - bc \neq 0$ ; here $ad - bc = -2$ . If $ad - bc$ were $0$ , the numerator would be a constant multiple of the denominator and the graph would collapse to a horizontal line.

Reduction to standard form

Dividing the numerator by the denominator rewrites the function.

y = \frac{(x-1) + 2}{x-1} = 1 + \frac{2}{x-1}

The graph is therefore the reciprocal $y = \dfrac{2}{x}$ translated one unit in each direction, a rectangular hyperbola. This rewriting shows why the graph of every linear fractional function is a translated reciprocal.

Domain, range and asymptotes

Item	Value
Domain	$x \neq 1$
Range	$y \neq 1$
Vertical asymptote	$x = 1$
Horizontal asymptote	$y = 1$
Centre	$(1, 1)$

Because $f(1+t) = 1 + \dfrac{2}{t}$ and $f(1-t) = 1 - \dfrac{2}{t}$ add up to $2$ , the graph has rotational symmetry about that centre. The two asymptotes meet at a right angle, which is what makes this a rectangular hyperbola.

Monotonicity and concavity

The derivative is $f'(x) = -\dfrac{2}{(x-1)^2}$ , negative throughout the domain, so the function decreases on each branch separately and has no extrema. The second derivative is $f''(x) = \dfrac{4}{(x-1)^3}$ , so the curve is concave up for $x > 1$ and concave down for $x < 1$ .

Sample points

$x$	$y$
$-3$	$\dfrac{1}{2}$
$-1$	$0$
$0$	$-1$
$2$	$3$
$3$	$2$

Inverse function

Solve $y = \dfrac{x+1}{x-1}$ for $x$ . Multiplying by $x-1$ and collecting terms gives $x(y-1) = y+1$ , hence $x = \dfrac{y+1}{y-1}$ . The inverse is the very same formula, so $f(f(x)) = x$ : a function that returns to the start when applied twice is called an involution. The fact that the curve passes through both $(2, 3)$ and $(3, 2)$ is a visible consequence, since the graph is symmetric about the line $y = x$ .

Applications

Linear fractional functions appear whenever two quantities are linked by a near-reciprocal relation. Solving the thin lens equation $\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{f}$ for the image distance gives $b = \dfrac{af}{a-f}$ , and the vertical asymptote is exactly the moment when an object placed at the focal point sends its image off to infinity.

Extended to complex numbers the same formula is a Mobius transformation¹, used in complex analysis and geometry because it maps circles and lines to circles and lines.

Mobius transformation, Wikipedia

Graph of the Linear Fractional Function y=x+1x−1y = \dfrac{x+1}{x-1}y=x−1x+1​