Reciprocal and rational functions

Maths: Analysis and Approaches · Functions. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

The reciprocal function f(x) = 1/x has a graph in two separate branches, in the first and third quadrants. It never touches the axes: x = 0 and y = 0 are its asymptotes. It is self-inverse, which means f⁻¹(x) = f(x), so applying it twice returns the starting number.

A rational function of the form (ax + b)/(cx + d) has the same two-branch shape, moved and stretched. An asymptote is a line the curve gets closer and closer to. The vertical one is where the denominator is zero. The horizontal one is the value y approaches when x is very large.

Find the two asymptotes and the two intercepts and you can sketch the graph.

Rules to remember

For y = (ax + b)/(cx + d): vertical asymptote where cx + d = 0, so x = −d/c
Horizontal asymptote: y = a/c (the ratio of the coefficients of x)
x-intercept: numerator = 0, so x = −b/a. y-intercept: put x = 0, so y = b/d
Domain: x ≠ −d/c. Range: y ≠ a/c
y = 1/x: asymptotes x = 0 and y = 0; domain x ≠ 0; range y ≠ 0
Self-inverse: f(f(x)) = x, and the graph is symmetric in the line y = x

Common mistake

Swapping the two asymptotes. The vertical asymptote comes from the denominator being zero (x = ...); the horizontal one comes from the coefficients of x (y = a/c).

Worked example

For y = (4x − 6)/(2x + 1), find the asymptotes and the intercepts with the axes.

Vertical asymptote: 2x + 1 = 0, so x = −1/2.
Horizontal asymptote: y = 4/2 = 2.
x-intercept: 4x − 6 = 0, so x = 3/2. y-intercept: y = (0 − 6)/(0 + 1) = −6.

Answer: Asymptotes x = −1/2 and y = 2; intercepts (3/2, 0) and (0, −6)

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Topic 2: Functions

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