Advanced Functions

1. Introduction

Once you are comfortable with polynomials, the next frontier is the family of functions built by dividing and rooting them: rational functions (ratios of polynomials) and radical functions (involving square roots, cube roots, and beyond). These functions are where the graph stops being a single smooth curve and starts doing dramatic things — shooting off to infinity near forbidden inputs, leveling out along invisible guide lines, and abruptly beginning or ending where a square root runs out of legal inputs.

Rational and radical functions are not just abstract exercises. They model real situations with built-in limits and trade-offs: average cost per unit (which spikes when production is tiny), the intensity of light or gravity falling off with distance, concentration in a mixture, and the time-versus-speed relationships of physics. Their characteristic features — asymptotes, holes, and domain restrictions — are precisely what make them able to describe behavior that "blows up" or "settles down."

In this article we learn to read a rational function like a map: factor it, find where it is undefined, distinguish a true vertical asymptote from a removable hole, and determine end behavior from a simple degree comparison. We then turn to radical functions, where the central skill is solving domain inequalities and handling the squaring step that can sneak in extraneous solutions. Throughout, the guiding question is always the same: what inputs are allowed, and what does the function do near the edges?

This material sits at the heart of Pre-Calculus: it connects the polynomial algebra you already know to the limit and asymptotic thinking you will need in calculus. Mastering asymptote analysis now means that when you later study limits formally, rational functions will already feel familiar rather than foreign.

2. Core Concepts

2.1 Rational functions and their domain

A rational function is a quotient of two polynomials,

$f(x) = \frac{p(x)}{q(x)}, \qquad q(x) \neq 0.$

Because division by zero is undefined, the domain is all real numbers except the zeros of the denominator. Finding the domain therefore begins with solving $q(x) = 0$ and excluding those values. For $f(x) = \dfrac{x+1}{x^2 - x - 6} = \dfrac{x+1}{(x-3)(x+2)}$, the excluded inputs are $x = 3$ and $x = -2$.

2.2 Vertical asymptotes versus holes

Not every excluded input behaves the same way. After factoring numerator and denominator completely, look at each zero of the original denominator:

• If a factor $(x - c)$ appears in the denominator only (or to a higher power there), the function grows without bound near $x = c$: this is a vertical asymptote, the line $x = c$.
• If a factor $(x - c)$ appears in both numerator and denominator and cancels completely, the function is undefined at $x = c$ but does not blow up: this is a hole (removable discontinuity) at $x = c$.

For example, in $\dfrac{x^2 - 1}{x - 1} = \dfrac{(x-1)(x+1)}{x-1}$ the factor $(x-1)$ cancels, leaving $x + 1$ with a hole at $x = 1$ — no asymptote. But $\dfrac{x+1}{x-1}$ has a genuine vertical asymptote at $x = 1$ because nothing cancels.

2.3 Deriving the hole location

After cancelling common factors, the simplified function $g(x)$ equals $f(x)$ everywhere except at the cancelled zeros. At $x = c$ where $(x-c)$ cancelled, the hole is at $\big(c,\, g(c)\big)$ — plug $c$ into the simplified form to find the $y$-coordinate. For $\dfrac{x^2-1}{x-1}$, the simplified form is $x+1$, so the hole is at $(1, 2)$.

2.4 End behavior and horizontal/slant asymptotes

Far out on the graph (as $x \to \pm\infty$), a rational function is controlled by the leading terms of $p$ and $q$. Why? Divide numerator and denominator by the highest power of $x$; all lower-degree terms vanish as $x \to \infty$, leaving only the ratio of leading coefficients when degrees match.

Compare the degree of the numerator, $\deg p$, with the degree of the denominator, $\deg q$:

• If $\deg p < \deg q$: the denominator outgrows the numerator, so $f(x) \to 0$. The horizontal asymptote is $y = 0$.
• If $\deg p = \deg q$: the function approaches the ratio of the leading coefficients. The horizontal asymptote is $y = \dfrac{a_n}{b_n}$.
• If $\deg p = \deg q + 1$: there is no horizontal asymptote but a slant (oblique) asymptote, found by polynomial long division — the quotient (ignoring the remainder) is the line.
• If $\deg p > \deg q + 1$: the end behavior follows a higher-degree polynomial (no linear asymptote).

A subtle point: a graph may cross a horizontal asymptote in the middle, even though it cannot cross a vertical one. The asymptote describes only the long-run behavior.

2.5 Intercepts of rational functions

The $x$-intercepts occur where the (simplified) numerator is zero, since a fraction equals zero only when its top is zero — but the value must still be in the domain. The $y$-intercept is simply $f(0)$, provided $0$ is in the domain. These anchor points, together with the asymptotes, let you sketch the whole graph.

2.6 Sign analysis near asymptotes

Between vertical asymptotes, a rational function maintains constant sign. To sketch, test one value in each interval determined by zeros of the numerator and denominator. For $f(x) = \dfrac{x+1}{x-1}$, test $x = 0$: $f(0) = -1 < 0$, so the graph is below the $x$-axis near zero.

Build a sign chart by listing all critical values (numerator zeros and denominator zeros) on a number line, then testing one sample point in each interval. This same technique applies to rational inequalities like $\dfrac{x+1}{x-1} > 0$, whose solution is the union of intervals where the sign chart shows positive.

2.7 Radical functions and domain inequalities

A radical function involves a root of a variable expression, such as $f(x) = \sqrt{x - 2}$ or $g(x) = \sqrt[3]{x+1}$. The key distinction is the parity of the root:

• Even roots (square root, fourth root, …) require a non-negative radicand, because an even root of a negative number is not real. The domain comes from solving the inequality $\text{radicand} \ge 0$.
• Odd roots (cube root, fifth root, …) accept all real inputs, since you can take the cube root of any number, positive or negative.

So $\sqrt{x-2}$ has domain $x \ge 2$, while $\sqrt[3]{x-2}$ has domain all real numbers.

2.8 Radicals in denominators

When an even root sits in a denominator, the radicand must be strictly positive ($> 0$), not merely non-negative, because the denominator cannot be zero. For $f(x) = \dfrac{1}{\sqrt{x-4}}$, we need $x - 4 > 0$, so $x > 4$.

2.9 Transformations of radical functions

The basic curve $y = \sqrt{x}$ starts at the origin and rises gently. Shifts and stretches move it predictably: $y = \sqrt{x - h} + k$ starts at the point $(h, k)$, and a negative inside, like $\sqrt{h - x}$, reflects it to open leftward. Recognizing the starting point and direction makes graphing radicals quick, and it foreshadows the general theory of function transformations.

2.10 Extraneous solutions

Solving radical equations almost always involves squaring both sides to remove the root. Squaring is not reversible — it can create solutions that satisfy the squared equation but not the original (because squaring erases sign information; both $3$ and $-3$ square to $9$). Such false answers are called extraneous, and you must check every candidate in the original equation and discard the ones that fail.

2.11 Rational equations

A rational equation contains variable expressions in denominators. The strategy: find excluded values, multiply both sides by the LCD to clear denominators, solve the resulting polynomial, and reject any solution that equals an excluded value.

2.12 Average cost as a rational function

If fixed costs are $F$ and variable cost per unit is $v$, then producing $x$ units costs $C(x) = F + vx$, and the average cost is

$A(x) = \frac{C(x)}{x} = \frac{F}{x} + v.$

As $x \to 0^+$, the $\dfrac{F}{x}$ term blows up — average cost spikes for small production runs. As $x \to \infty$, $A(x) \to v$, the horizontal asymptote.

2.13 Inverse variation

When one quantity is inversely proportional to another, their relationship is rational: $y = \dfrac{k}{x}$ for constant $k$. The graph is a hyperbola with vertical asymptote $x = 0$ and horizontal asymptote $y = 0$. Gravitational force and illumination intensity both follow inverse-square laws: $F \propto \dfrac{1}{r^2}$, a rational function of distance $r$.

2.14 Nested radicals

Expressions like $\sqrt{x + \sqrt{x}}$ require working from the inside out. The inner radical $\sqrt{x}$ needs $x \ge 0$; the outer radical then needs $x + \sqrt{x} \ge 0$, which is automatically satisfied when $x \ge 0$. Domain: $x \ge 0$. Nested radicals appear in some geometry formulas and in continued-radical puzzles.