What's f(x, -x^2, x^2)?

Question

If f(x, y, z)=(z^2+x^2-y)/(2x^2+y), what's f(x, -x^2, x^2)? Answer: x^2+2

I have no idea to do this.

Answer 1

The information that is given is: f(x, y, z)=(z^2+x^2-y)/(2x^2+y) AND f(x, -x^2, x^2). This is saying that f is a function that is dependent on variables x, y, and z and is defined as (z² + x² - y)/(2x² + y). Furthermore the variables y and z are defined in terms of the variable x, so that y= -x² and that z= x². Thus: x = x, y = -x², and z = x².

(z² + x² - y)/(2x² + y) and x = x, y = -x², and z = x² _____ given

{(x²)² + x² - (-x²)}/{2x² + (-x²)} _____ substituting for y and z

(x⁴ + x² + x²)/(2x² - x²) _____ simplifying exponents and signs

(x⁴ + 2x²)/(x²) _____ combining liked terms

(x²)(x² + 2)/(x²) _____ factoring x² in the numerator

x² + 2 _____ canceling x² from numerator and denominator [ANSWER]

 

Answer 2

Hi Sun,

F(x, y, z) is called function notation, it is really just a substitution problem.  It tells you what to substitute for x, y, and z.  In your problem, f (x, -x^2, x^2) means x does not change, y is replaced with -x^2, and z is replaced with x^2.

So substitute:  ( (x^2)^2 + x^2 - (-x^2) )/(2x^2 + (-x^2) )  then clear parentheses...

(x^4 + 2x^2)/x^2  after using the rules of exponents, signed numbers and combining like terms.

then x^4/x^2 = x^2 and 2x^2/x^2 = 2, hence your solution x^2 + 2