The pH of a sample of lake water was originally 6 when measured in 1987. When measured again in 2014, the pH of the water was 4. How has the concentration of hydrogen ions in the lake changed?

1. The concentration of hydrogen ions has increased by a factor of 2.
2. The concentration of hydrogen ions has increased to 10 times its original value.
3. The concentration of hydrogen ions has increased to 100 times its original value.
4. The concentration of hydrogen ions has decreased by a factor of 2.

kepler used the data from tycho blahe (his mentor) to construct the laws of planetary motion.

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .