Consider the function
f (x)=-3x^5+9x^3+2x, determine if f is an even or odd function and what type of symmetry would occur. Justify your reasoning. (2 marks)
To determine if a function is even or odd, we need to check its symmetry with respect to the y-axis (even) and the origin (odd).
1. Even function:
A function f(x) is even if and only if f(x) = f(-x) for all x in the domain.
2. Odd function:
A function f(x) is odd if and only if f(x) = -f(-x) for all x in the domain.
Now, let's apply these conditions to the given function f(x) = -3x^5 + 9x^3 + 2x:
1. Even function test:
f(x) = -3x^5 + 9x^3 + 2x
f(-x) = -3(-x)^5 + 9(-x)^3 + 2(-x)
= -3(-x^5) + 9(-x^3) - 2x
= -3(-x^5 + 9x^3 - 2x)
= -3x^5 + 9x^3 - 2x
Since f(x) = f(-x), the function passes the even function test.
2. Odd function test:
f(x) = -3x^5 + 9x^3 + 2x
-f(-x) = -[-3(-x)^5 + 9(-x)^3 + 2(-x)]
= -[-3(-x^5) + 9(-x^3) - 2x]
= 3x^5 - 9x^3 + 2x
Since -f(-x) is not equal to f(x), the function fails the odd function test.
Therefore, the function f(x) = -3x^5 + 9x^3 + 2x is an even function. This means it exhibits symmetry with respect to the y-axis.