Step 1 :The question is asking for a monomial function that tends to infinity as x tends to both positive and negative infinity. This means that the function must be an even function, because even functions are symmetric about the y-axis and thus have the same behavior as x tends to positive and negative infinity. Therefore, we are looking for a function with an even exponent.
Step 2 :The functions \(f(x)=-2 x^{2}\), \(f(x)=x^{8}\), and \(f(x)=\left(\frac{1}{5}\right) x^{4}\) are even functions, which means they could be the function g described in the question. However, we also need to check if these functions tend to infinity as x tends to both positive and negative infinity.
Step 3 :The functions \(f(x)=x^{8}\) and \(f(x)=\left(\frac{1}{5}\right) x^{4}\) are even functions that tend to infinity as x tends to both positive and negative infinity. Therefore, these could be the function g described in the question.
Step 4 :The possible function formulas for \(f\) are \(f(x)=x^{8}\) and \(f(x)=\left(\frac{1}{5}\right) x^{4}\). So, the final answer is \(\boxed{f(x)=x^{8}, f(x)=\left(\frac{1}{5}\right) x^{4}}\).