A one-to-one function is a type of mathematical function in which each input (x-value) has a unique output (y-value). This means that no two different inputs can produce the same output. It is also known as an injective function.
To prove that a function is one-to-one, you must show that for any two distinct inputs, the corresponding outputs are also distinct. This can be done algebraically by setting the two inputs equal to each other and solving for the variable. If the outputs are not equal, then the function is one-to-one.
A strictly increasing interval is a range of values for which the function is increasing, meaning that the output values are getting larger as the input values increase. In other words, the slope of the function is positive within this interval.
To determine the strictly increasing intervals of a function, you must find the intervals where the function's derivative (slope) is positive. This can be done by finding the critical points (where the derivative is equal to zero or undefined) and testing points on either side of these points to see if the derivative is positive.
Yes, a function can be both one-to-one and strictly increasing. In fact, all one-to-one functions are also strictly increasing, as each input has a unique output. However, not all strictly increasing functions are one-to-one, as there may be different inputs that produce the same output.