Transcendental (Non-Algebraic) Functions: Concepts & Procedures

Examples of Transcendental Expressions. Transcendental (also called non-algebraic) expressions are expressions that cannot be constructed solely from polynomials by adding, subtracting, multiplying, dividing, raising to an integer power, or taking roots. The list includes some algebraic-looking operations such as: powers of x that are irrational numbers like x^π, exponential functions like 3^x, and tower functions like x^x. Most transcendental expressions have special names.

Examples of Transcendental Functions. The list includes special function names such as: log(x), tan(x), arctan(x), tanh(x), and arctanh(x). Functions that must be expressed using a piecewise definition are also transcendental.

Periodic Transcendental Functions. A periodic function is one whose graph repeats over fixed interval lengths. The six basic trigonometric functions—sin(x), cos(x), tan(x), csc(x), sec(x), and cot(x)—all have this property. Here is an example of the sine function with three different arguments: x, 2x, and x/2.

Non-periodic Transcendental functions. Among the functions studied in calculus, only the trigonometric functions are periodic. So, in addition to irrational powers and tower functions, the list of basic, transcendental functions includes exponential (e^x) and logarithmic (log(x)) functions, the six inverse trigonometric functions (arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), and arccot(x)), the six hyperbolic functions (sinh(x), cosh(x), tanh(x), csch(x), sech(x), and coth(x)), and the six inverse hyperbolic trigonometric functions (arcsinh(x), arccosh(x), arctanh(x), arccsch(x), arcsech(x), and arccoth(x)). Here is an example of the exponential functions 2^x, 3^x, and 4^x. They all go through the point (0,1) and have a horizontal asymptote, y=0.

Complete Graphs for NonPeriodic Functions. A complete graph is one that includes all the important mathematical features. For nonperiodic transcendental functions, this means the viewing window should be just large enough to display all x- and y-intercepts, all maxima and minima, all points of inflections, all holes, all jumps (1- and 2-hole), all vertical asymptotes, all horizontal asymptotes, all vertical tangents, all vertical cusps, all corners, and all isolated points. When you enter a nonperiodic transcendental function and click the GraphIt button, WebGraphing.com automatically displays a complete graph. Armed with this knowledge, you can select different viewing windows to zoom in on particular aspects of the graph or refine it more to your liking.

Fixed Graph Intervals for Periodic Functions. Periodic functions have infinitely-many points of mathematical interest. All the basic trig functions have infinitely-many x-intercepts, all but tan(x) and cot(x) have infinitely-many local maxima and local minima, all have infinitely-many points of inflection, and all but sin(x) and cos(x) have infinitely-many vertical asymptotes. For such functions, it is impossible to show all the important mathematical features in a graph since the features continue beyond the bounds of any display region. Consequently, whenever a trigonometric function is graphed on WebGraphing.com, we restrict the display of the graph to the interval from −2π to 2π. You can subsequently graph again over any interval of your own choosing to explore the graph or refine it more to your liking.

Zoom In/Out (Resizing the Viewing Window). If you wish to graph again with a different display window, you can do this in two ways: (1) you can enter the lower and upper x-bounds only and let WebGraphing.com determine optimal y-bounds (Smart Zooming) that include all the important mathematical features contained within your x-bounds, or (2) you can enter both the lower and upper x- and y-bounds to choose the precise viewing window.  The first option is especially useful if you are zooming in on a peak or valley or a change in concavity, since in that case WebGraphing.com chooses an optimal range of y-values.  The choice of an appropriately-chosen display can be important to visually verify, beyond the indicated color-coding, underlying features like maxima, minima, and points of inflection.

Domain

Domain.An irrational number power, like x^π, is only defined for non-negative values of x. Thus, the domain of such functions consists of nonnegative numbers. Other transcendental functions may have restrictions to their domains resulting from their precise definitions.

Horizontal and Vertical Asymptotes
(What are Asymptotes?)

Horizontal Asymptotes. A horizontal asymptote is a horizontal line, y=c, which the graph of a function approaches either as x tends to +∞and/or as x tends to −∞. This is customarily indicated on the graph by plotting the horizontal asymptote as a dashed horizontal line. Unlike vertical asymptotes, the curve representing the function may touch or cross a horizontal asymptote. Transcendental functions may have up to two horizontal asymptotes.

Vertical Asymptotes. A vertical asymptote is a vertical line, x=c, which the graph of a transcendental function approaches. As x approaches the number c through domain values from either the left or right side, the y-values of the function (from at least one side) tend to either +∞ or −∞. This is customarily indicated on the graph by plotting the vertical asymptote as a dashed vertical line. The value x=c of a vertical asymptote is sometimes referred to as an essential singularity, since no redefinition of the function at that x-value can ever make it continuous (for contrast, see holes below). Using a piecewise defined function, it is possible, thru redefinition at the vertical asymptote, for a transcendental function to touch a vertical asymptote once. See the next graph for an example.

Holes and Jump Discontinuities

Holes. With transcendental functions, a hole can occur at a point along with a vertical asymptote, as happens with the exponential function e^(1/x). In this example, the function is nearly continuous but only from one side.

Jump Discontinuities. If, at a given point, a function has limits from the left side and right side that are finite but unequal, the function is said to have a jump discontinuity at that point. Here is an example of a transcendental function that, at x=0, has a left hand limit of – π/2 and a right hand limit of π/2

Vertical Tangent Lines, Vertical Cusps, and Corners

Maxima and Minima

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