Exponential and power functions defns.

First of all, this has nothing to do with HW help or anything, jsut want to clear some things up. I had a few quick questions in terms of what actually is considered a power function.

Would x^2-25 be considered a power function, with k=1, and p=2? Would the -25 just be considered the y-intercept then? Or a power funtion HAVE to be exactly in kx^p form?

Also,(4+x)/(5x^4) is only a polynomial, and is technically not an exponential or power function correct?

This brings me to my final question: Is a power function considered a polynomial? But is a polynomial function considered a power function?

Thanks for any input.

Protip 1: clean up your text, you're being sloppy.  x^2-25 is an expression, not a function. Then later on you say k=1 and p=2, but you've never mentioned k or p before!  People reading your post are thinking "WTF are k and p?"

Answer 1: f(x)=x^2-25 is NOT a power function.  Power functions are { x |-> k*x^a | a in R, k in R }, and usually where x in R+.  If take a power function and add or subtract a constant, you violate the scale invariance that power functions share.  The scale invariance is that f(c * x) is proportional to f(x) for constant c.

Answer 2: The function f(x)=(4+x)/(5x^4) is neither polynomial, exponential, or a power function.  It is, however, a rational function.

Answer 3: Power functions are polynomial if and only if the exponent is a non-negative integer.  So f(x)=5x^4 is a power function and a polynomial, but f(x)=3x^2.75 and f(x)=x^-1 are power functions but not polynomials.

Protip 2: In mathematics, stay away from "technically" and "considered".  Either X is a Y or it isn't... because in math, all our definitions are exact.  It's worth mentioning because no other field of study has exact definitions.  The times it's worth hedging are when people disagree on definitions.  For example, "Some authors require rings to have multiplicative identities."  After this you'd say whether or not you require rings to have multiplicative identities.