To determine which equations represent inverse variations, let's first understand what an inverse variation is. Inverse variation implies that as one variable increases, the other variable decreases, and their product remains constant. Mathematically, this is typically expressed as \(y = \frac{k}{x}\) where \(k\) is a non-zero constant.
Let’s analyze each of the given functions to see if they fit the form of an inverse variation:
Equation I: \(f(x) = \frac{-1}{2x}\)
- This equation is in the form \(y = \frac{k}{x}\), where \(k = -\frac{1}{2}\).
- Hence, this is an example of an inverse variation.
Equation II: \(f(x) = 3x\)
- This equation represents a direct variation, where the output \(f(x)\) changes directly in proportion to \(x\).
- It does not fit the form \(y = \frac{k}{x}\).
- Therefore, this is not an example of inverse variation.
Equation III: \(f(x) = \frac{-z}{xy}\)
- Here, we need to consider the variable relationships. If we assume \(z\) to be a constant, then \(f(x)\) varies inversely with the product \(xy\).
- This can be viewed as: \(f(x) = \frac{K}{x}\), where \(K = -\frac{z}{y}\) and \(K\) is treated as a constant for a constant \(y\).
- Thus, this represents inverse variation as well.
From our analysis:
- Equation I fits inverse variation.
- Equation II does not fit inverse variation.
- Equation III fits inverse variation.
So, among the given equations, I and III are examples of inverse variation.
The correct choice is:
C. I and III only.
