key term - Increasing

Definition

Increasing refers to the state or process of becoming greater in size, amount, or degree over time. In the context of functions and function notation, increasing describes a relationship where the output values of a function grow larger as the input values become greater.

5 Must Know Facts For Your Next Test

1. An increasing function is one where the output values consistently grow larger as the input values increase.
2. Increasing functions have a positive rate of change, meaning the function's values rise as the input increases.
3. Increasing functions can be linear, quadratic, exponential, or take on other functional forms.
4. The domain of an increasing function must be a set of real numbers, as the input values need to be ordered.
5. Increasing functions are often used to model real-world phenomena where a quantity grows as another quantity increases, such as the relationship between time and distance traveled.

Review Questions

• Explain how the concept of increasing relates to the domain and range of a function.
  • The domain of an increasing function must be a set of ordered real numbers, as the input values need to be able to be compared and ordered from smaller to larger. The range of an increasing function will also be a set of ordered real numbers, as the output values will consistently grow larger as the inputs increase. The relationship between the domain and range is critical for understanding increasing functions, as the input and output values must align with the increasing pattern.
• Describe the key characteristics of the graph of an increasing function.
  • The graph of an increasing function will have a positive slope, meaning it rises from left to right. The function's values will consistently grow larger as you move from left to right along the graph. Increasing functions may take on various shapes, such as linear, quadratic, or exponential, but they will all share the common characteristic of having a graph that rises. The steepness of the graph can indicate the rate of increase, with steeper graphs representing faster rates of increase.
• Analyze how the concept of increasing functions can be applied to model real-world phenomena.
  • Increasing functions are commonly used to model situations where a quantity grows as another quantity increases. For example, the relationship between time and distance traveled for a vehicle traveling at a constant speed is an increasing function, as the distance traveled will consistently grow larger as time passes. Similarly, the relationship between the number of hours worked and the total earnings for an employee is an increasing function, as wages earned will increase with more hours worked. Understanding increasing functions is crucial for analyzing and predicting the behavior of these types of real-world relationships.

Related terms

Function:

A function is a relation between a set of inputs and a set of permissible outputs, where each input is paired with exactly one output.

Domain:

The domain of a function is the set of all possible input values for the function.

Range:

The range of a function is the set of all possible output values for the function.