Questions: Observe the behavior of f(x^2) = -x^2(x+1). Whether each function is even, odd.

Transcript text: Observe the behavior of $f\left(x^{2}\right)=-x^{2}(x+1)$. Whether each function is even, odd.

Solution

Solution Steps

Step 1: Analyze the function

The given function is $f(x) = -x^2(x+1)$. We can expand this to $f(x) = -x^3 - x^2$.

Step 2: Test for even function

A function is even if $f(-x) = f(x)$. Let's find $f(-x)$: $f(-x) = -(-x)^3 - (-x)^2 = -(-x^3) - x^2 = x^3 - x^2$. Since $x^3 - x^2 \neq -x^3 - x^2$, the function is not even.

Step 3: Test for odd function

A function is odd if $f(-x) = -f(x)$. We already found $f(-x) = x^3 - x^2$. Now let's find $-f(x)$: $-f(x) = -(-x^3 - x^2) = x^3 + x^2$. Since $x^3 - x^2 \neq x^3 + x^2$, the function is not odd.