Summary of Basic Classes of Functions

The power function [latex]f(x)=x^n[/latex] is an even function if [latex]n[/latex] is even and [latex]n \ne 0[/latex], and it is an odd function if [latex]n[/latex] is odd.
The root function [latex]f(x)=x^{1/n}[/latex] has the domain [latex][0,\infty )[/latex] if [latex]n[/latex] is even and the domain [latex](-\infty,\infty )[/latex] if [latex]n[/latex] is odd. If [latex]n[/latex] is odd, then [latex]f(x)=x^{1/n}[/latex] is an odd function.
The domain of the rational function [latex]f(x)=p(x)/q(x)[/latex], where [latex]p(x)[/latex] and [latex]q(x)[/latex] are polynomial functions, is the set of [latex]x[/latex] such that [latex]q(x) \ne 0[/latex].
Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.
A polynomial function [latex]f[/latex] with degree [latex]n \ge 1[/latex] satisfies [latex]f(x) \to \pm \infty[/latex] as [latex]x \to \pm \infty[/latex]. The sign of the output as [latex]x \to \infty[/latex] depends on the sign of the leading coefficient only and on whether [latex]n[/latex] is even or odd.
Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the [latex]x[/latex]– and [latex]y[/latex]-axes are examples of transformations of functions.

Key Equations

Point-slope equation of a line
[latex]y-y_1=m(x-x_1)[/latex]
Slope-intercept form of a line
[latex]y=mx+b[/latex]
Standard form of a line
[latex]ax+by=c[/latex]
Polynomial function
[latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+ \ldots +a_1x+a_0[/latex]

algebraic function: a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable [latex]x[/latex]

cubic function: a polynomial of degree 3; that is, a function of the form [latex]f(x)=ax^3+bx^2+cx+d[/latex], where [latex]a \ne 0[/latex]

degree: for a polynomial function, the value of the largest exponent of any term

linear function: a function that can be written in the form [latex]f(x)=mx+b[/latex]

logarithmic function: a function of the form [latex]f(x)=\log_b(x)[/latex] for some base [latex]b>0, \, b \ne 1[/latex] such that [latex]y=\log_b(x)[/latex] if and only if [latex]b^y=x[/latex]

mathematical model: A method of simulating real-life situations with mathematical equations

piecewise-defined function: a function that is defined differently on different parts of its domain

point-slope equation: equation of a linear function indicating its slope and a point on the graph of the function

polynomial function: a function of the form [latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+ \cdots +a_1x+a_0[/latex]

power function: a function of the form [latex]f(x)=x^n[/latex] for any positive integer [latex]n \ge 1[/latex]

quadratic function: a polynomial of degree 2; that is, a function of the form [latex]f(x)=ax^2+bx+c[/latex] where [latex]a \ne 0[/latex]

rational function: a function of the form [latex]f(x)=p(x)/q(x)[/latex], where [latex]p(x)[/latex] and [latex]q(x)[/latex] are polynomials

root function: a function of the form [latex]f(x)=x^{1/n}[/latex] for any integer [latex]n \ge 2[/latex]

slope: the change in [latex]y[/latex] for each unit change in [latex]x[/latex]

slope-intercept form: equation of a linear function indicating its slope and [latex]y[/latex]-intercept

standard form: equation of a linear function with both variable terms set equal to a constant, [latex]ax+by=c[/latex].

transcendental function: a function that cannot be expressed by a combination of basic arithmetic operations