Answer
$h(x)=1/f(x)$ is a 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 $$\frac{1}{f(x_1)}\ne \frac{1}{f(x_2)}\hspace{1cm}\text{whenever}\hspace{1cm}x_1\ne x_2$$
Now consider $h(x)$, since $h(x)=1/f(x)$, that means
$$h(x_1)=\frac{1}{f(x_1)}\hspace{1cm}\text{and}\hspace{1cm}g(x_2)=\frac{1}{f(x_2)}$$
However, as we proved above $1/f(x_1)\ne 1/f(x_2)$, hence:
$$h(x_1)\ne h(x_2)\hspace{1cm}\text{whenever}\hspace{1cm}x_1\ne x_2$$
According to the definition of one-to-one functions, $h(x)$ is a one-to-one function.
