Algebraically, determine whether the following function is even, odd, or neither. Make sure to include your work for full credit. f(x)=x3−2x+3 (1 point) Responses Odd Odd Neither Neither Even Even
To determine whether the function \( f(x) = x^3 - 2x + 3 \) is even, odd, or neither, we check the symmetry of the function about the y-axis (even) or the origin (odd).
For a function to be even, it must satisfy \( f(x) = f(-x) \) for all \( x \) in its domain.
For a function to be odd, it must satisfy \( f(-x) = -f(x) \) for all \( x \) in its domain.
Let’s analyze \( f(x) \) by replacing \( x \) with \( -x \) to evaluate \( f(-x) \):
\( f(-x) = (-x)^3 - 2(-x) + 3 \)
\( f(-x) = -x^3 + 2x + 3 \)
Now let's compare this to \( f(x) \) and \( -f(x) \):
\( f(x) = x^3 - 2x + 3 \)
\( -f(x) = -x^3 + 2x - 3 \)
We can see that \( f(-x) \) is not the same as \( f(x) \), which means the function is not even.
\( f(-x) = -x^3 + 2x + 3 \) is also not the negative of \( f(x) \), \( -f(x) = -x^3 + 2x - 3 \); hence, the function is not odd.
Since \( f(x) \) is neither even nor odd, the correct answer is:
Neither