Power, Polynomial, Rational, and Algebraic Functions: Key Concepts and Rules

Power function

A function of the form f(x)=cx^r, where c and r are constants.

What makes a function a power function?

The variable x is the base and is raised to a fixed power.

Rational number

A number that can be written as m/n, where m,n are integers and n≠0.

Rational power

A power q that can be written as m/n.

Zero exponent

x^0=1 for all x.

Positive integer exponent

x^n means multiplying x by itself n times.

Odd root

x^(1/n)=ⁿ√x when n is a positive odd integer.

Even root

x^(1/n)=ⁿ√x, but x must be ≥0.

Positive rational exponent

x^(m/n)=(x^(1/n))^m=(x^m)^(1/n).

Negative exponent

x^(-m/n)=1/x^(m/n).

Negative exponents rule

a^(-b)=1/a^b.

Product rule

a^b·a^c=a^(b+c).

Quotient rule

a^b/a^c=a^(b-c).

Power of a product

(ab)^c=a^c·b^c.

Power of a quotient

(a/b)^c=a^c/b^c.

Power of a power

(a^b)^c=a^(bc), assuming the required conditions hold.

Radical product rule

ⁿ√(ab)=ⁿ√a·ⁿ√b.

Radical quotient rule

ⁿ√(a/b)=ⁿ√a/ⁿ√b.

Radical power rule

ⁿ√(a^m)=(ⁿ√a)^m=a^(m/n).

Polynomial

A function written as a_nx^n+...+a_1x+a_0.

Polynomial exponent restriction

The powers of x must be nonnegative whole numbers.

Term of a polynomial

Each individual expression a_kx^k in the polynomial.

Coefficient

The number multiplying a power of x.

Constant term

The term with x^0; usually written as a_0.

Leading term

The term with the highest power of x.

Leading coefficient

The coefficient of the leading term.

Degree

The highest power of x with a nonzero coefficient.

Constant polynomial

A polynomial of degree 0.

Linear polynomial

A polynomial of degree 1.

Quadratic polynomial

A polynomial of degree 2.

Cubic polynomial

A polynomial of degree 3.

Implied domain of a polynomial

All real numbers.

Rational function

A quotient of two polynomials: f(x)=p(x)/q(x).

Implied domain of a rational function

All real x for which q(x)≠0.

How do you find a rational function's domain?

Set the denominator ≠0 and solve for x.

Algebraic function

A function formed using arithmetic operations and rational powers of x.

Four basic algebraic function types

Linear, polynomial, rational, and power functions.

Can a function belong to multiple types?

Yes. Function classifications can overlap.

Is x² a power function?

Yes. It has the form cx^q with c=1 and q=2.

Is x² a polynomial?

Yes. It is a polynomial of degree 2.

Is 1/x a power function?

Yes. 1/x=x^(-1).

Is 1/x a rational function?

Yes. It is 1 divided by the polynomial x.

Transcendental function

A function that is not algebraic.

Examples of transcendental functions

Exponential, logarithmic, trigonometric, and inverse trigonometric functions.

How do you identify a power function?

Rewrite it in the form Ax^k.

How do you identify a polynomial?

Check that every x exponent is a nonnegative whole number.

How do you identify a rational function?

Check whether it can be written as one polynomial divided by another.

How do you identify an algebraic function?

Check whether it uses arithmetic operations and rational powers.

Can a rational function have a zero denominator?

No. Those x-values are excluded from its domain.

Fraction multiplication

(a/b)(c/d)=ac/bd.

Adding fractions with like denominators

a/c+b/c=(a+b)/c.

Adding fractions with unlike denominators

Find a common denominator, then add the numerators.

Multiplying by 1

You can multiply a fraction by c/c without changing its value.