What are Power Functions? | Functions and Relations, Types of Functions

What are power functions? Power functions are functions of the form f(x) = kx^a where k and a are both constant real numbers. So, in a power function, the variable is raised to a power. Other definitions restrict the coefficient k to being equal to 1, but that definition seems less common and does not describe a family of functions as useful in reality. We go over the definition of power functions and some examples and interesting properties in today's full math video lesson!

All power functions with positive even exponents have a parabolic shape, and if they have a coefficient of 1 they will intersect the points (1,1) and (-1,1). There are lots of interesting properties of power functions depending on their exponents and coefficients, try to figure some out on your own, and why the properties exist!

You also might notice, if you know what odd and even functions are, that power functions with even exponents are even functions, and power functions with odd exponents are odd functions! That's where "odd" and "even" functions get their name!

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.

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+WRATH OF MATH+

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[rod_katt]

Thank you! But if my level of English was a little higher, I would be able to understand more information (hello, from Russia, where I cannot find a normal lesson). So thanks again for this video.

[aalbeed]

Hi
I have a question about power functions

There are some people say that
" a power function is any function of the form f(x) = k x^a, where k and a are nonzero constant real numbers."

My question is which definition is correct?
1) all real numbers
2) all real numbers except for zero



Thank you so much

[alexcozar]

Hi, very informative, thank you for the clear explanation. I have a couple of questions. In the curve that you mention seems to have asymptotes (the black one), what would the K value indicate? Correct me if I am wrong, would it be the middle point of the curve? And, how could I calculate the x value at which we reach the asymptote in the y axis? I will appreciate If you could suggest some references available in the internet too. Thanks!

[davec]

I have put log10-log10 transformed data onto linear axes and plotted as a linear relationship y = a + bx. I want to also plot this relationship as a power function on linear axes, and it appears that to do this I take the coefficients from the previous relationship and plot y = 10^a * x^b. The 10^a makes sense but I'm not quite sure where the x^b comes from. Any insights would be appreciated.

[mlzd]

you much. You're really good at this!

[ndDayRiffs]

Thank you. My online class sucks. It's not even a joke. It's a bad joke.

[bethelyemane]

No that is not correct power function is a function defined as kx^a, where K is d/t from 0 and a is a rational number.

[anileh]

His voice reminds me of sheldon from the big bang theory