Algebraic Functions: Concepts & Procedures

Examples of Algebraic Expressions. Algebraic expressions are constructed from polynomials by adding, subtracting, multiplying, dividing, raising to an integer power, and taking roots. Examples include: x^1/2, 3 x^5/3 – 17, –x^–1 + 7, and x²– x^–2– 4.

Algebraic functions. Algebraic functions are expressions that involve only the algebraic operations of addition, subtraction, multiplication, division, raising to an integer power, and extracting an odd number root or even number root. Strictly speaking, any polynomial or rational function is also an algebraic function.

The absolute value function, abs(x)=|x|, is of special interest. Since it can also be expressed as (x²)^1/2, it is an algebraic function.

n^th Roots of a number. An n^th root (n is a positive integer) of a given number is a number that, when raised to the n^th power (that is, when n copies of an n^th root are multiplied), produces the given number. There are n distinct n^th roots (DeMoivre's Theorem) of any nonzero number, and they may be real or imaginary numbers. No matter how large n is, at most two of the n^th roots are real, and when there are two real roots, they are of opposite sign.

Odd roots of a number. If n ≥ 3 is odd, there is exactly one real root of a given (real) number. For example, the three cube roots of –1 are: –1 and (1/2)(1±3^1/2i), where i is the imaginary unit, i=(–1)^1/2. When graphing functions with odd roots in the x-y plane, plotting is restricted to the real-valued y-coordinates. Below are the graphs of the cube root, fifth root, and seventh root functions. All go through the points (–1,1), (1,1) and the origin, (0,0). They are defined for all values of x.

Even roots of a number. If n ≥ 2 is even and a given (real) number is positive, there are two real n^th roots of the number, numerically the same but opposite in sign. For example, the four 4^th roots of 1 are ±1 and ±i . If n ≥ 2 is even and a given (real) number is negative, all n^th roots are imaginary. For example, the four 4^th roots of –1 are (2^1/2/2)(±1±i). Below are the graphs of the square root, fourth root, and sixth root functions. All go through the points (1,1) and the origin, (0,0). They are only real-valued for nonnegative values of x. For that reason, their graphs only appear in the right half-plane x≥0.

Complete Graph. A complete graph is one that includes all the important mathematical features. For algebraic 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 2-hole jumps, all vertical asymptotes, all horizontal asymptotes, all vertical tangents, all vertical cusps, all corners, and all isolated points. When you enter an algebraic 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.

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 and Intercepts

Domain. Algebraic functions involving even roots may require restrictions to their domains in order to produce real y-values. For example, (x–1)^1/2 is real valued only when x ≥ 1. Also, any x-value producing zero in the denominator of an algebraic function must be excluded, as in ((1–x)^1/2)/x, where x≠0 since division by 0 is not defined. The domain of an algebraic function is the largest set of real x-values that produce real y-values.

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 algebraic function may touch or cross a horizontal asymptote. While rational functions may have at most one horizontal asymptote, algebraic functions may have up to two horizontal asymptotes.

Vertical Asymptotes. A vertical asymptote is a vertical line, x=c, which the graph of an algebraic function approaches but never touches. For algebraic functions, vertical asymptotes always occur at the zeros of the denominator that are not eliminated by improper cancellation. As x approaches such a denominator zero through domain values from either one of the left or right sides, the y-values of the function tend to either +∞ or −∞. This is customarily indicated on the graph by plotting the vertical asymptote as a dashed vertical line. The curve never intersects the dashed vertical line, since the defining algebraic function does not have any y-value at the denominator zero. 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).

Holes and Jump Discontinuities

Holes. When an algebraic function has a common factor in the numerator and denominator that can be eliminated in the denominator by improper cancellation, the importance of this lies in the fact that the algebraic function is not defined at the zero of the common factor since it appears in the denominator. Any common factor that is eliminated in the denominator due to improper cancellation does not give rise to a vertical asymptote. This is traditionally signified on the graph by having an empty circle appear at the one or two places where the function might otherwise be continuous from at least one side. When the limit from the left is finite and equals the limit from the right, this is called a hole and is sometimes referred to as a removable singularity (or removable discontinuity), since, unlike a vertical asymptote, the algebraic function could be redefined to be continuous at that point.

Jump Discontinuities. When an algebraic function is undefined at a point, say, for example x/|x| at x=0, but the limits from the left side and right side are finite but unequal (in this example, as x tends to 0 from the left, x/|x| tends to –1; as x tends to 0 from the right, x/|x| tends to +1), this gives rise to a jump discontinuity. This is traditionally signified on the graph by having an empty circle appear at the two places where the function has limits. Here is an example of an algebraic function with a jump discontinuity.

Vertical Tangent Lines, Vertical Cusps, and Corners

Vertical Tangents and Vertical Cusps. If a function f is differentiable at x=a, then the graph of f has a tangent line at a. What happens when f is not differentiable? The graph of y=f (x) is said to have a vertical tangent line at (a, f (a)) if: (1) f is continuous at a, (2) f '(a) does not exist (as a real number), and (3) f '(x) approaches one of either ∞ or −∞ as x approaches a. An example of this is y=x^1/3 at (0,0). Note that if x=a is an endpoint of an interval in a restricted domain, f is said to have a vertical tangent as determined from one side only; in that case, the continuity of f and the limit of f '(x) are interpreted as x approaches a from one side only. An example of this is y=x^1/2 at when a=0. As x approaches 0 from the right, f '(x) approaches ∞. Geometrically, the secant lines through (0,0) and (x,x^1/2) approach vertical as x approaches 0. In the case where a is an interior point of the domain, if the approach of f '(x) is +∞ as x approaches a from one side and −∞ as x approaches a from the other side, a sharp point occurs that is called a vertical cusp. An example of this is y=|x|^1/3 at (0,0).

Corners. The graph of y=f (x) is said to have a corner at (a, f (a)) if: (1) f  is continuous at a, (2) f '(a) does not exist, and (3) f '(x) approaches a real number as x approaches a from the left side and f '(x) approaches a different real number as x approaches a from the right side. An example of this is the absolute value function, abs(x)=|x|, at (0,0), where f '(x) approaches –1 as x approaches 0 from the left side and f '(x) approaches +1 as x approaches 0 from the right side.

Maxima and Minima

Local Maxima. The y-value f (c) is a local maximum value (also called relative maximum value) of f if there is an open interval containing the x-value c (half-open if c is an endpoint in the domain of f ) where f (c) ≥ f (x). For interior points in the domain of f, the First Derivative Test states that when the graph of the function f, continuous at x=c, is increasing on the immediate left of the number x=c and decreasing on the immediate right of the number x=c, then the value of f at c is locally the largest, i.e., f (c) is a local maximum. This test can be extended to endpoints in the domain of f: if x=c is a left endpoint and decreasing on the immediate right, or if x=c is a right endpoint and increasing on the immediate left, then f (c) is a local maximum.

Local Minima. The y-value f (c) is a local minimum value (also called relative minimum value) of f if there is an open interval containing the x-value c (half-open if c is an endpoint in the domain of f ) where f (c) ≤ f (x). For interior points in the domain of f, the First Derivative Test states that when the graph of the function f, continuous at x=c, is decreasing on the immediate left of the number x=c and increasing on the immediate right of the number x=c, then the value of f at c is locally the smallest, i.e., f (c) is a local minimum. This test can be extended to endpoints in the domain of f: if x=c is a left endpoint and increasing on the immediate right, or if x=c is a right endpoint and decreasing on the immediate left, then f (c) is a local minimum.

Determining Local Maxima and Local Minima.Any value of x in the domain of f is called a critical number of f ' (also called critical point or critical value) if either f '(x)=0 or f '(x) does not exist. For continuous functions, the local maxima and local minima can only occur at the critical numbers or endpoints of the domain of f. These numbers separate the domain of f into intervals. To determine all local maxima and local minima, select a convenient "Test Value" on each such interval and determine the sign of f '(Test Value). At each critical number or endpoint, there are three possibilities.

1. If the sign of f '(Test Value) is positive (+) on the left side (interval to the left) of a critical number of f ' and negative (−) on the right side, suggesting visually that the function is increasing on the left side and decreasing on the right side, then f  has a local maximum at that critical number. If the sign of f '(Test Value) is positive (+) on the left side of a right endpoint (suggesting visually that the function is increasing on the left side) or negative (−) on the right side of a left endpoint (suggesting visually that the function is decreasing on the left side), then f has a local maximum at that endpoint.

2. If the sign of f '(Test Value) is negative (−) on the left side of a critical number of f ' and positive (+) on the right side, suggesting visually that the function is decreasing on the left side and increasing on the right side, then f has a local minimum at that critical number. If the sign of f '(Test Value) is negative (−) on the left side of a right endpoint (suggesting visually that the function is decreasing on the left) or negative (+) on the right side of a left endpoint (suggesting visually that the function is increasing on the right side), then f has a local minimum at that endpoint.

3. If there is no change in sign of f '(Test Value) from either side of a critical number to the other side, then the critical number is not a local maximum or local minimum for f.

Increasing. Geometrically speaking, a function f is increasing on an interval if the function is rising over the interval as you look at the graph from left to right. By definition, f is increasing on an interval I if f (a) < f (b) whenever a < b, for a, b in I. This is true if f '(x) > 0 at interior points of the interval I. The latter condition is adequate (sufficient) to establish that a function is increasing, but it does not always work to verify the “increasing” property, as indicated by the example of the algebraic function f (x)=x^1/3, where f ' does not exist at x=0 but y=f (x) is increasing on any interval containing 0. The increasing segments (also called arcs) of a graph are shown on WebGraphing.com in red.

Decreasing. Geometrically speaking, a function y=f (x) is decreasing on an interval if the function is falling over the interval as you look at the graph from left to right. By definition, y=f (x) is decreasing on an interval I if f (a) > f (b) whenever a < b, for a, b in I. This is true if f '(x) < 0 at interior points of the interval I. The latter condition is adequate (sufficient) to establish that a function is decreasing, but it does not always work to verify the “decreasing” property, as indicated by the example of the algebraic function f (x)=–x^1/3, where f ' does not exist at x=0 but y=f (x) is decreasing on any interval containing 0. The decreasing segments of a graph are shown on WebGraphing.com in blue.

Determining Increasing and Decreasing Intervals. To determine the intervals of increase and decrease for an algebraic function f, first find all the critical numbers of f. These numbers separate the domain of f into intervals. Select a convenient Test Value on each such interval and determine the sign of f '(Test Value). If the sign is positive (+), then the function is increasing on the interval. If the sign is negative (−), then the function is decreasing on the interval. Based on these considerations, the intervals of increase and decrease can be determined by creating a table with critical numbers of f and test values.

Points of Inflection. A point on a graph where the function is continuous and there is a change in concavity is called a point of inflection. Any point where the function is not continuous cannot be a point of inflection.

Determining Points of Inflection. Any value of x in the domain of f is called a critical number of f '' if either f ''(x)=0 or f ''(x) does not exist. For any continuous function, a point of inflection can only occur at a real-valued zero of f ''. For continuous functions, the points of inflection can only occur at the critical numbers in the domain of f. These numbers separate the domain of f into intervals. To determine all points of inflection, select a convenient "Test Value" on each such interval and determine the sign of f ''(Test Value). At each critical number of f '', there are three possibilities:

1. If the sign of f ''(Test Value) is positive (+) on the left side (interval to the left) of a critical number of f '' and negative (−) on the right side (interval to the right), indicating visually that the function is concave up on the left side and concave down on the right side, then f has a point of inflection at that critical number.

2. If the sign of f ''(Test Value) is negative (−) on the left side of a critical number of f '' and positive (+) on the right side, indicating visually that the function is concave down on the left side and concave up on the right side, then f has a point of inflection at that critical number.

3. If there is no change in sign of f ''(Test Value) from one side of a critical number of f '' to the other side, then the zero is not a point of inflection for f.

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