Rational and Modulus Functions: Graphs and Examples

A rational function is a fraction where both the numerator and the denominator are polynomials, written as f(x) = p(x) / q(x) where q(x) is not the zero polynomial. The modulus function, written as f(x) = |x|, gives the distance of a number from zero, so it is always non-negative: |−7| = 7 and |7| = 7.

These two functions matter because they introduce shapes that simple straight lines and parabolas cannot make. Rational functions produce curves with asymptotes — lines the graph approaches but never touches — which is how real relationships like speed versus time for a fixed distance behave. The modulus handles expressions that can never go negative, a skill reused in error and distance problems.

Key Concepts

• Rational function: f(x) = p(x)/q(x) with p, q polynomials and q(x) ≠ 0. Its domain excludes every x where q(x) = 0.
• Vertical asymptote: a vertical line x = a where the denominator is zero and the numerator is not; the graph shoots up or down toward it.
• Horizontal asymptote: a horizontal line y = b the graph settles toward as x → ∞ or x → −∞.
• Modulus function: |x| = x if x ≥ 0, and |x| = −x if x < 0. Its graph is a V with the corner at the origin.

Formulas

• General rational function: f(x) = (a_nx^n + … + a_0) / (b_mx^m + … + b_0), defined only where the denominator ≠ 0.
• Simple hyperbola: f(x) = 1/x has vertical asymptote x = 0 and horizontal asymptote y = 0. Units depend on the application; for speed = distance/time the unit is m/s.
• Modulus: |x| = x for x ≥ 0; |x| = −x for x < 0. Also, |x − a| measures the distance between x and a on the number line, so |x − a| < d means a − d < x < a + d.

Worked Example

Sketch and analyse f(x) = (2x + 1) / (x − 3).

1. Domain: the denominator is zero when x − 3 = 0, i.e. x = 3. So the domain is all real numbers except x = 3.
2. Vertical asymptote: x = 3, because the numerator at x = 3 is 2(3) + 1 = 7 ≠ 0.
3. Horizontal asymptote: both degrees are 1, so y = 2/1 = 2.
4. x-intercept: set numerator = 0: 2x + 1 = 0 gives x = −1/2.
5. y-intercept: f(0) = 1/(−3) = −1/3.

Common Misconceptions

• "The graph crosses the vertical asymptote." It never does. An asymptote is a wall the function can only approach; near x = 3 the values of the example become huge in magnitude, positive on one side and negative on the other.
• "|x| is differentiable everywhere like x²." No — the modulus has a sharp corner at x = 0, so its slope jumps from −1 to +1 there and the derivative does not exist at the corner.
• "Cancelling factors removes asymptotes." If numerator and denominator share a factor like (x − 2), cancelling gives a hole at x = 2, not an asymptote — a point is missing from the graph, while the curve does not blow up there.

Real-World Applications

• Average cost: cost per item = (fixed cost + variable cost) / (number of items) is a rational function; the average cost approaches the variable cost per item as production grows.
• Speed and travel: for a fixed journey, speed = distance / time is rational in time; halving the time doubles the required speed.
• Error bounds: expressions like |measured − true| use the modulus to state measurement error, and tolerances such as |x − 5| < 0.1 appear in manufacturing quality control.

Rational functions are polynomial fractions with asymptotes at the zeros of the denominator, while the modulus function graphs as a V measuring distance from zero. Together they are your first functions with restricted domains, broken graphs and corners.