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Power Functions


The basic power functions are

Power Functions: positive integer exponent | matematicasVisuales

n is called the exponent. We start studying power functions with positive integer exponent. The expression xn is known as 'x to the nth power'.

This family includes lines, parabolas, cubic parabolas, etc.

They are the basis of the polynomials.

They are examples of even and odd functions. The y-axis of an even function is a symmetry axis for the function: one half of the graph is a 'mirror image' of the other half. When the exponent is even, the function is even:

Power Functions: Even function | matematicasVisuales

Some functions are symmetric with respect to the origin. These functions are called odd functions. When the exponent of a power functions is odd, the function is odd:

Power Functions: Odd function | matematicasVisuales

Even and odd power functions have a different end behavior (to the right and to the left).

Seeing this applet we can (intuitively) accept these limits:

Power Functions: power functions limit | matematicasVisuales

Power Functions: power function limit | matematicasVisuales

The inverse of exponentiation is extracting a root (the nth root functions):

Power Functions: root functions | matematicasVisuales

Root functions are power functions with exponent the reciprocal of a positive integer (n is called the root index).

One function and its inversa are simmetrical respect the first quadrant diagonal. You get the graph of the inverse function reflecting the graph across the line y=x .

Power Functions: root function with index an even positive integer | matematicasVisuales
Power Functions: root function with index an odd positive integer | matematicasVisuales

The domain of these functions is all the real numbers when n is odd and only the non-negative real numbers when n is even.

Playing with the mathlet you can accept (intuitively) this limit:

Power Functions: root functions limit | matematicasVisuales

The next step is considering power functions with positive rational exponent:

Power Functions: positive rational exponent | matematicasVisuales

REFERENCES

Richard Courant y Fritz John - Introducción al cálculo y al análisis matemático. Ed. Limusa-Wiley.

MORE LINKS

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