To determine which statement is always true, let's evaluate each option based on mathematical principles involving increasing and decreasing functions.
- Option 1: \(f(x)\) is decreasing, then \(\frac{1}{f(x)}\) is increasing.
Explanation: If a function \(f(x)\) is decreasing, it means that for any two points \(x_1\) and \(x_2\) where \(x_1 < x_2\), we have \(f(x_1) > f(x_2)\). The statement that \(\frac{1}{f(x)}\) is increasing is not always true. For \(\frac{1}{f(x)}\) to be increasing, \(f(x)\) must be decreasing and positive. But if \(f(x)\) crosses zero or becomes negative, this won't hold. Therefore, this statement is not always true.
- Option 2: \(f(x)\) is decreasing, then \(\frac{1}{f(x)}\) is also decreasing.
Explanation: As established in option 1, for the reciprocal function \(\frac{1}{f(x)}\), if \(f(x)\) is decreasing and positive, \(\frac{1}{f(x)}\) is increasing. Thus, this statement is also not always true.
- Option 3: If both \(f\) and \(g\) are positive functions such that \(f\) is decreasing and \(g\) is increasing, then \(\frac{f}{g}\) is a decreasing function.
Explanation: Given two positive functions where \(f(x)\) is decreasing and \(g(x)\) is increasing, the function \(\frac{f}{g}\) will indeed be decreasing. This is because the numerator is getting smaller and the denominator is getting larger, both contributing to the reduction of the overall fraction value. Therefore, this statement is always true.
- Option 4: If both \(f\) and \(g\) are positive functions such that \(f\) is increasing and \(g\) is decreasing, then \(\frac{f}{g}\) is a decreasing function.
Explanation: If \(f(x)\) is increasing and \(g(x)\) is decreasing, then \(\frac{f}{g}\) may not be decreasing. An increasing numerator and decreasing denominator would typically cause the overall value of the fraction to increase rather than decrease. Thus, this statement is not always true.
Conclusion: The statement that is always true is: If both \(f\) and \(g\) are positive functions such that \(f\) is decreasing and \(g\) is increasing, then \(\frac{f}{g}\) is a decreasing function.
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