Types of Functions III

Logarithmic and piecewise functions are two essential types of mathematical functions.

A logarithmic function is the inverse of an exponential function and reflects across the line y equals x. Exponential functions start slowly and then grow rapidly, while logarithmic functions rise quickly at first and then slow down.

This behavior allows large numerical differences to be represented more conveniently using logarithmic scaling.

For example, the pH scale, which measures acidity or basicity, uses a logarithmic scale.

Each unit on the pH scale represents a tenfold change in hydrogen ion concentration.

On the other hand, a piecewise function uses different rules for different sections of its graph, and these rules can change suddenly depending on the input.

For example, a taxi fare might increase in fixed steps as the distance travelled increases.

The graph of a piecewise function looks like separate line segments or curves, with each segment following its own rule.

Piecewise functions are useful for modelling situations involving jumps, breaks, or varying behaviour across different input ranges.