Rational (Reciprocal) Functions and Power Functions

Your group hires a bus for a trip. It costs 120 euros and you split the price equally. If 4 of you go, each pays 30 euros. If 8 go, each pays only 15 euros. The more people, the less each one pays, but the price never drops to zero. This is inverse proportion y = 120/x, and its graph is a hyperbola.

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The hyperbola and its asymptotes

The function y = k/x (k ≠ 0) is not defined for x = 0 — you cannot divide by zero. Its graph has two branches.

  • For x close to zero, k/x is a huge number: the graph "shoots off" up or down along the y-axis.
  • For large x, k/x is almost zero: the graph "snuggles up" to the x-axis.

Lines that the graph keeps getting closer to but never reaches are called asymptotes. For y = k/x they are the x-axis and the y-axis.

  • k > 0: the branches are in the first and third quadrants, and the function decreases on each branch.
  • k < 0: the branches are in the second and fourth quadrants, and the function increases on each branch.

A shifted hyperbola

The dashed hyperbola y = 1/x and the orange hyperbola y = 3/(x − 2) + 1 with asymptotes x = 2 and y = 1

The function

y =

has the same hyperbola as y = k/x, just shifted: m to the right and n up. The asymptotes are:

  • vertical x = m — where the denominator is zero,
  • horizontal y = n — the number that y gets close to for large x.

Worked example 1

y = .

  1. The denominator x − 2 = 0 for x = 2. Vertical asymptote x = 2.
  2. For large x the fraction 3/(x − 2) is almost zero, so y ≈ 1. Horizontal asymptote y = 1.
  3. The domain is ℝ ∖ {2}, the range is ℝ ∖ {1}.

⚠️ Watch the sign: y = 5/(x + 3) − 2 has the asymptotes x = −3 and y = −2.

The general form of a rational function

The rational function we study here is a quotient of two linear expressions:

y = , where c ≠ 0 and the numerator is not a multiple of the denominator.

We find the asymptotes by dividing the fraction — rewriting it in the form k/(x − m) + n.

Worked example 2

y = .

  1. Write the numerator using the denominator: 2x + 1 = 2(x − 1) + 3.
  2. y = .
  3. Asymptotes: x = 1 and y = 2.

💡 Shortcut: the vertical asymptote is where the denominator is zero. The horizontal one is the quotient of the numbers in front of x: y = a/c. Here 2/1 = 2.

Power functions

A power function has the formula y = xⁿ. Its properties depend on whether the exponent n is even or odd and whether it is positive or negative.

The graphs of y = x², y = x⁴ and y = x⁻²; all are symmetric about the y-axis Even positive exponent (x², x⁴, …): the graph looks like a parabola, the function is even, the domain is ℝ, the range is [0, ∞), it has a minimum of 0, and it increases on [0, ∞). Even negative exponent (x⁻² = 1/x², x⁻⁴, …): the function is even, the domain is ℝ ∖ {0}, the range is (0, ∞), it decreases on (0, ∞), and it has no minimum. The graphs of y = x³ and y = x⁻¹; both are symmetric about the origin Odd positive exponent (x³, x⁵, …): the function is odd and increasing on the whole of ℝ; the domain and the range are both ℝ. Odd negative exponent (x⁻¹ = 1/x, x⁻³, …): a hyperbola, the function is odd, the domain and the range are both ℝ ∖ {0}, and it decreases on (0, ∞).

We read x⁻² as "x to the power of minus two", and it means 1/x².

💡 All graphs y = xⁿ pass through the point (1, 1). For positive n they also pass through (0, 0).

Common mistakes

  • Wrong sign of the asymptote. For 1/(x + 4) the vertical asymptote is x = −4.
  • Forgetting the horizontal asymptote. For y = k/(x − m) + n it is y = n, not the x-axis.
  • x⁻² = −x². No! x⁻² = 1/x², and it is always positive.
  • "The hyperbola is decreasing." For k > 0 it decreases on each branch separately, but it is not decreasing on its whole domain: −1 < 1, and yet 1/(−1) < 1/1.

Summary

  • y = k/(x − m) + n: a hyperbola with asymptotes x = m and y = n.
  • (ax + b)/(cx + d): divide, or use x = −d/c and y = a/c.
  • y = xⁿ: even n gives an even function, odd n gives an odd function.
  • Negative n: the graph never touches the axes, and the domain does not contain zero.

Try it yourself

  1. Find the asymptotes of y = 4/(x + 1) − 3.
  2. Find the asymptotes of y = (x + 5)/(x − 2).
  3. Is the function y = x⁻³ even or odd? What is its domain?
Answers:
  1. x = −1, y = −3.
  2. x + 5 = (x − 2) + 7, so y = 1 + 7/(x − 2): x = 2, y = 1.
  3. Odd (the exponent −3 is odd), and the domain is ℝ ∖ {0}.

Practice

👉 Next article: One-to-One and Inverse Functions