Final answer:
To determine f(-x), substitute -x for each x in the given function and apply the power of a power rule to simplify. The signs of terms with odd exponents will change, resulting in f(-x) = 6x⁵ + 9x⁴ - 6x³ - 6x² - 36.
Step-by-step explanation:
To find f(-x) for the given function f(x) = -6x⁵ + 9x⁴ + 6x³ - 6x² - 36, we need to substitute -x for every instance of x in the function and simplify.
- Replace every x with -x.
- Apply the power of a power rule: (-x)^n = (-1)^n × x^n.
- Simplify the terms
The function f(-x) will become:
f(-x) = -6(-x)⁵ + 9(-x)⁴ + 6(-x)³ - 6(-x)² - 36
Now, calculate the powers:
f(-x) = -6(-1)⁵×x⁵ + 9(-1)⁴×x⁴ + 6(-1)³×x³ - 6(-1)²×x² - 36
f(-x) = -6(-1)x⁵ + 9(1)x⁴ + 6(-1)x³ - 6(1)x² - 36
f(-x) = 6x⁵ + 9x⁴ - 6x³ - 6x² - 36
As you can see, the sign of the term changes with odd exponents when substituting -x.