What is a rational function?
A rational function is f(x) = P(x)/Q(x), where P and Q are polynomials and Q is not the zero polynomial. "Rational" comes from "ratio".
Simple examples: y = 1/x, y = 3/(x + 2), y = (x² − 1)/(x − 3).
Domain: all real numbers except where Q(x) = 0. You can never divide by zero.
y = k/x is called inverse proportion: when x doubles, y halves. Its graph is a hyperbola with two branches.
Asymptotes: vertical, horizontal and slant
An asymptote is a line the graph gets closer and closer to.
Vertical asymptote (VA)
At x = c where the bottom is 0 but the top is not. Near it, y goes to +∞ or −∞.
Horizontal asymptote (HA)
Compare degrees of top (n) and bottom (m):
- n < m → y = 0
- n = m → y = (leading coefficient of top) ÷ (leading coefficient of bottom)
- n = m + 1 → no HA, but a slant (oblique) asymptote; divide to find it.
A graph may cross a horizontal asymptote in the middle; it can never cross a vertical one.
Holes, intercepts and sketching step by step
If a factor cancels from top and bottom, the graph has a hole (a missing point), not an asymptote. Example: (x² − 4)/(x − 2) = x + 2 with a hole at (2, 4).
- Factor top and bottom. Cancel common factors → holes.
- Domain: bottom ≠ 0.
- VA from the remaining bottom zeros.
- HA or slant asymptote from degrees.
- x-intercepts: top = 0. y-intercept: put x = 0.
- Test a point on each side of every VA to see if the graph goes up or down. Then sketch.
Rate of change: the graph gets steeper near a VA and flatter near the HA.
Reciprocal and modulus graphs, transformations and inverses
Reciprocal graph y = 1/f(x): where f(x) = 0 you get a VA; where f is large, 1/f is small; f and 1/f have the same sign; they meet where f = ±1.
Modulus: y = |f(x)| reflects the negative part up; for y = 1/|x| both branches sit above the x-axis.
Transformations: y = a/(x − h) + k moves 1/x right by h, up by k, and stretches by a. VA x = h, HA y = k.
Inverse: for y = (ax + b)/(cx + d), swap x and y and solve: f⁻¹(x) = (b − dx)/(cx − a). The VA and HA swap.
Derivative (quotient rule): (P/Q)′ = (P′Q − PQ′)/Q². For y = 1/x, y′ = −1/x², always negative, so each branch falls.
Key formulas and definitions
- f(x) = P(x)/Q(x), Q(x) ≠ 0
- VA: x = c where Q(c) = 0 and P(c) ≠ 0 (after cancelling)
- HA: deg P < deg Q → y = 0; equal → y = ratio of leading coefficients
- Slant asymptote when deg P = deg Q + 1 (divide)
- y = a/(x − h) + k: VA x = h, HA y = k
- Inverse of (ax + b)/(cx + d) is (b − dx)/(cx − a)
- Quotient rule: (P/Q)′ = (P′Q − PQ′)/Q²
Worked examples
1. Find the domain of f(x) = 5/(x + 4).
Bottom = 0 when x = −4. Domain: all real x except −4.
2. Find the asymptotes of y = 3/(x − 2) + 1.
It is 3/x moved 2 right and 1 up. VA x = 2, HA y = 1.
3. For y = (2x + 1)/(x − 1), find VA, HA and intercepts.
VA: x − 1 = 0 → x = 1. HA: degrees equal, 2/1 → y = 2. x-intercept: 2x + 1 = 0 → x = −½. y-intercept: (0 + 1)/(0 − 1) = −1.
4. Find the hole of y = (x² − 9)/(x − 3).
x² − 9 = (x − 3)(x + 3). Cancel (x − 3): y = x + 3, x ≠ 3. Hole at (3, 6). No VA.
5. Find the slant asymptote of y = (x² + 1)/x.
Divide: (x² + 1)/x = x + 1/x. As x grows, 1/x → 0, so the slant asymptote is y = x. VA x = 0.
6. Find the inverse of f(x) = (x + 2)/(x − 3) and its asymptotes.
y = (x + 2)/(x − 3). Swap: x = (y + 2)/(y − 3) → xy − 3x = y + 2 → y(x − 1) = 3x + 2 → f⁻¹(x) = (3x + 2)/(x − 1). f has VA x = 3, HA y = 1; f⁻¹ has VA x = 1, HA y = 3 (swapped).
7. Sketch y = 1/(x² − 4).
Bottom = (x − 2)(x + 2), so VA x = −2 and x = 2. Degree top 0 < 2, HA y = 0. y-intercept −¼, no x-intercept. Test x = 3: +1/5 (above). x = 0: −¼ (below). x = −3: +1/5. So the middle piece is an upside-down cup below the axis, and the outer pieces sit above, falling toward y = 0.
Common mistakes
- Forgetting to cancel common factors first, and calling a hole an asymptote.
- Putting the VA where the top is zero. The top being zero gives an x-intercept.
- Thinking a graph can never cross a horizontal asymptote. It can, in the middle part.
- Using the constant terms instead of leading coefficients for the HA, e.g. saying y = 1/−1 for (2x + 1)/(x − 1).