Use algebra to determine whether the function 𝑓(𝑥) = 22𝑥3 − 19𝑥 is even, odd, orneither.

1. Break Down the Problem

To determine whether the function $f(x) = 22x^3 - 19x$ is even, odd, or neither, we need to evaluate $f(-x)$ and compare it to $f(x)$ and $-f(x)$.

2. Relevant Concepts

• A function is even if $f(-x) = f(x)$ for all $x$ in the domain.
• A function is odd if $f(-x) = -f(x)$ for all $x$ in the domain.

3. Analysis and Detail

1. Compute $f(-x)$: $f(-x) = 22(-x)^3 - 19(-x)$ Since $(-x)^3 = -x^3$, we have: $f(-x) = 22(-x^3) + 19x = -22x^3 + 19x$

2. Compare $f(-x)$ with $f(x)$ and $-f(x)$:

   • We know $f(x) = 22x^3 - 19x$.
   • Compute $-f(x)$: $-f(x) = - (22x^3 - 19x) = -22x^3 + 19x$

3. Compare:

   • $f(-x) = -22x^3 + 19x$
   • $-f(x) = -22x^3 + 19x$

4. Verify and Summarize

Since $f(-x) = -f(x)$, this confirms that the function is odd.

Final Answer

The function $f(x) = 22x^3 - 19x$ is odd.