Answer
$g(x)=-f(x)$ is an one-to-one function.
Work Step by Step
If $f(x)$ is one to one, according to defintition, that means $$f(x_1)\ne f (x_2)\hspace{1cm}\text{whenever}\hspace{1cm}x_1\ne x_2$$
So we can deduce that $$-f(x_1)\ne -f(x_2)\hspace{1cm}\text{whenever}\hspace{1cm}x_1\ne x_2$$
Now consider $g(x)$, since $g(x)=-f(x)$, that means
$$g(x_1)=-f(x_1)\hspace{1cm}\text{and}\hspace{1cm}g(x_2)=-f(x_2)$$
However, as we proved above $-f(x_1)\ne -f(x_2)$, hence:
$$g(x_1)\ne g(x_2)\hspace{1cm}\text{whenever}\hspace{1cm}x_1\ne x_2$$
According to the definition of one-to-one functions, $g(x)$ is an one-to-one function.
