If f(x)= (4cos^4x+2cos^2x+2sin^2x-x^9-1/2cos4x)^(1/9). Find f(f(2))
To find f(f(2)), we need to first evaluate f(2) and then substitute the result into the function f(x) again.
Let's start by finding f(2):
1. Rewrite the given function f(x): f(x) = (4cos^4x + 2cos^2x + 2sin^2x - x^9 - 1/2cos4x)^(1/9)
2. Substitute x = 2 into f(x): f(2) = (4cos^4(2) + 2cos^2(2) + 2sin^2(2) - 2^9 - 1/2cos4(2))^(1/9)
3. Simplify the expression inside the parentheses: f(2) = (4cos^4(2) + 2cos^2(2) + 2sin^2(2) - 512 - 1/2cos8)^(1/9)
Now, let's substitute f(2) into the function f(x) again:
4. Substitute x = f(2) into f(x): f(f(2)) = (4cos^4(f(2)) + 2cos^2(f(2)) + 2sin^2(f(2)) - f(2)^9 - 1/2cos4(f(2)))^(1/9)
Note that at this point, we cannot simplify the expression any further since f(2) is an unknown value. To obtain a numerical solution, f(2) needs to be evaluated using a calculator or software that can calculate trigonometric functions and powers accurately.
So, the final expression for f(f(2)) is (4cos^4(f(2)) + 2cos^2(f(2)) + 2sin^2(f(2)) - f(2)^9 - 1/2cos4(f(2)))^(1/9).
just plug and chug!
f(2) = -509.8^(1/9) = -1.999 ≈ -2
f(-2) = 514.1927^(1/9) = 2.0009 ≈ 2