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Power functions

Real functions of the form f (x) = c x n {\displaystyle f(x)=cx^{n}}, where c ≠ 0 {\displaystyle c\neq 0}, are sometimes called power functions. When n {\displaystyle n} is an integer and n ≥ 1 {\displaystyle n\geq 1}, two primary families exist: for n {\displaystyle n} even, and for n {\displaystyle n} odd. In general for c > 0 {\displaystyle c>0}, when n {\displaystyle n} is even f (x) = c x n {\displaystyle f(x)=cx^{n}} will tend towards positive infinity with increasing x {\displaystyle x}, and also towards positive infinity with decreasing x {\displaystyle x}. All graphs from the family of even power functions have the general shape of y = c x 2 {\displaystyle y=cx^{2}}, flattening more in the middle as n {\displaystyle n} increases. Functions with this kind of symmetry (f (− x) = f (x) {\displaystyle f(-x)=f(x)}) are called even functions.

When n {\displaystyle n} is odd, f (x) {\displaystyle f(x)} 's asymptotic behavior reverses from positive x {\displaystyle x} to negative x {\displaystyle x}. For c > 0 {\displaystyle c>0}, f (x) = c x n {\displaystyle f(x)=cx^{n}} will also tend towards positive infinity with increasing x {\displaystyle x}, but towards negative infinity with decreasing x {\displaystyle x}. All graphs from the family of odd power functions have the general shape of y = c x 3 {\displaystyle y=cx^{3}}, flattening more in the middle as n {\displaystyle n} increases and losing all flatness there in the straight line for n = 1 {\displaystyle n=1}. Functions with this kind of symmetry (f (− x) = − f (x) {\displaystyle f(-x)=-f(x)}) are called odd functions.

For c < 0 {\displaystyle c<0}, the opposite asymptotic behavior is true in each case.