Power Functions With Negative Exponents In Mathematics

Power Functions with negative exponents

d/dx(√x) = (1/2√x)d/dx(x^1/2) = (1/2x^-1/2)

A power function with a negative exponent is a function of the form f(x) = x^(-n), where n is a positive integer. This type of function can also be written as 1/x^n.

One important property of power functions with negative exponents is that they are undefined at x = 0. This is because division by zero is undefined, and f(0) = 1/0^n = 1/0, which is undefined.

In addition, as x approaches infinity, f(x) approaches 0. This is because as x becomes large, the denominator of the fraction 1/x^n becomes very large, causing the entire fraction to approach zero.

Similarly, as x approaches negative infinity, f(x) approaches 0 in the case where n is an even integer power; or the limit does not exist if n is an odd integer power.

Another important property is that power functions with negative exponents are always decreasing functions. This is because as x increases, the denominator of the fraction 1/x^n becomes larger, causing the entire fraction to become smaller, and vice versa as x decreases.

Finally, it is worth noting that power functions with negative exponents can be used to model situations in which a quantity decays over time or distance. For instance, radioactive decay can be modeled using a power function with a negative exponent, where the exponent represents the half-life of the radioactive substance.

More Answers: