How is the 'Answer' achieved in the 'Problem' after x,y-axis reflection?

Question

Problem: $f(x) = x^3 - x^2 + x – 1$ is reflected in both the $x$-axis and $y$-axis. What is the final equation to represent this transformation?

Answer: $x^3 + x^2 + x +1$

My solution: let $g(x)=-f(x)$ << reflection across $x$-axis and $h(x)$ be $g(-x)$ << reflection across $y$-axis.

I ended up having:

$g(x) = -x^3 +x^2 -x+1$ << multiplied every term in $f(x)$ by $-1$

$h(x) = g(-x) = -(-x^3 +x^2 -x)+1 $

MY FINAL ANSWER: $h(x)=x^3 +x^2 -x+1$ which is different from the equation at 'Answer', how do I achieve the equation in 'Answer'?

Answer 1

The first part you did was correct. However, given your

$$g(x) = -x^3+x^2-x+1 \tag{1}\label{eq1}$$

then the correct next part is

$$h(x) = g(-x) = -(-x)^3 + (-x)^2 - (-x) + 1 = x^3 + x^2 + x + 1 \tag{2}\label{eq2}$$

You mistake was taking the negative of the terms in the expression involving $x$, rather than substituting $x$ with $-x$ in the expression, as I did above.

Answer 2

$$-f(-x) = - [(-x)^{3} - (-x)^{2} + (-x) - 1 ] = - [-x^{3} - x^{2} - x - 1] = x^{3} + x^{2} + x + 1 $$