Function Definition?

Question

I just learning about this stuff so I need some help please.

Given f(x) = x^2 and x>0 then prove

f[SQRT(x) + SQRT(f(x))]
/ (aka divided by)
f(x+1) - f(x-1)

is a constant. (I think this means it does not depend on x.

An explaination would be very helpful, thank you very much.

May have typed it wrong... :( it is,

f(SQRT (x)) + SQRT (f(x))
/ (divided by)
f(x+1) - f(x-1)

Answer 1

You can't. Note that if g(x) = f(√x + √f(x)) / (f(x+1) - f(x-1)), g(1) = 1, but g(2) = (2+√2)²/8 ≈ 1.4571 ≠ 1. If you actually meant to write f(√x + √f(x)) /f(x+1) - f(x-1), this still would not be a constant, because if h(x) = f(√x + √f(x)) /f(x+1) - f(x-1), h(1) = 1, but h(2) = (2+√2)²/9 -1 ≈ 0.2952 ≠ 1. So either you typed in the formula wrong, or this is supposed to be a trick question, because the formula you gave is not constant.

Edit: Okay, the new formula looks right. In the future though, when the numerator and/or denominator have more than one term, please either surround them with parentheses or extend the fraction bar above them. For example, (a+b)/(c+d) is acceptable, as is:
a+b
------
c+d

but:

a+b
/
c+d

Will be interpreted as a+(b/c)+d, which is not what you want.

Moving on, you have:

(f(√x) + √f(x))/(f(x+1) - f(x-1))

Evaluate the functions:

((√x)² + √(x²))/((x+1)² - (x-1)²)

Now at this point we simplify. Note that the square root and square are inverse functions, so they will cancel each other out (technically, each number has two square roots, and since the √ function by convention refers only to the positive one, √(x²)=|x|, which may not equal x. However, it was already stated that x>0, so |x| = x and we can leave that consideration out. Therefore:

((√x)² + √(x²))/((x+1)² - (x-1)²)
(x+x)/((x+1)²-(x-1)²)

Expand the terms in the denominator:

(2x)/((x²+2x+1)-(x²-2x+1))

Cancel terms:

(2x)/(4x):

Simplify:

1/2

Note that this is a constant, and so we are done.

Answer 2

Don't books explain this?