Function Evaluation

[nycmath]

If f(x) = (4 + x^2)/(1 + x^4), find f(G(x)) when G(x) = 1 - x + x^3

[Eigenvalue]

f(g(x)) is using the g(x) as the input for f(x)
f(g(x))=[tex]\frac{4+(1-x+x³)²}{1+(1-x-x³)⁴}[/tex]
Simplify the numerator: 4+(1-x+x³)²=x⁶-2x⁴+2x³+x²-2x+5
Simplify the denominator: x¹²-4x¹⁰+4x⁹+6x⁸-12x⁷+2x⁶+12x⁵-11x⁴+6x²-4x
f(g(x))=[tex]\frac{x⁶-2x⁴+2x³+x²-2x+5}{x¹²-4x¹⁰+4x⁹+6x⁸-12x⁷+2x⁶+12x⁵-11x⁴+6x²-4x}[/tex]
It can be simplified, yet the result remains the same

[nycmath]

f(g(x)) is using the g(x) as the input for f(x)
f(g(x))=[tex]\frac{4+(1-x+x³)²}{1+(1-x-x³)⁴}[/tex]
Simplify the numerator: 4+(1-x+x³)²=x⁶-2x⁴+2x³+x²-2x+5
Simplify the denominator: x¹²-4x¹⁰+4x⁹+6x⁸-12x⁷+2x⁶+12x⁵-11x⁴+6x²-4x
f(g(x))=[tex]\frac{x⁶-2x⁴+2x³+x²-2x+5}{x¹²-4x¹⁰+4x⁹+6x⁸-12x⁷+2x⁶+12x⁵-11x⁴+6x²-4x}[/tex]
It can be simplified, yet the result remains the same

Wow! The end result is a monstrosity of a fraction. Cool.