There is not one set investment demand function in the sense that there is one Pythagorean Theorem or one equation for the money multiplier. Instead, the investment demand function is something that can change. It is simply the relationship between the interest rate and the amount of investment that is demanded.
This curve can shift for a variety of reasons and that means that the function can change when those factors change. In other words, we know that there is a relationship between the interest rate and the amount of investment but we do not know exactly what that relationship is because it varies as other factors vary.
Investment demand function refers to the relationship between level of investment and the cost of capital. The cost of capital is the real interest rate. This relationship is very important because interest rate is the major means available to governments to influence investment.
In taking decisions on investment in any projects, firms compare the the revenue from the project with the cost of capital, which depends on the rate of interest. The net of revenue less the cost of capital represents the profit from the project. Then this is positive the investment makes money, while a negative figure signifies losses. Firms will therefore invest in a project only when the revenues form the project is higher than the cost of capital. As the cost of interest, and with that cost of capital rises, there are less projects that meets this condition. Therefor investment falls with increase in interest rates, and rises with fall in interest rates. The exact relationship between the interest rate and demand for investment will depend on availability of opportunities for investing and their revenue potential.