Increasing Function

Definition

An increasing function is a type of mathematical function where, for any two points within its domain, if one point has a greater input value than the other, the corresponding output value is also greater. This characteristic means that as you move from left to right on the graph of the function, the y-values rise. Recognizing this property helps in understanding how functions behave under transformations and how to graph them accurately.

5 Must Know Facts For Your Next Test

1. A function is considered increasing on an interval if for any two points within that interval, the output values increase as the input values increase.
2. Graphically, an increasing function will show an upward slope when plotted on a coordinate plane.
3. Linear functions with a positive slope are examples of increasing functions.
4. In terms of transformations, shifting a graph vertically or horizontally does not change whether it is increasing.
5. For a function to be classified as strictly increasing, it must satisfy that for every pair of points in its domain, if the inputs are different, then the outputs must also be different.

Review Questions

• How can you determine if a function is increasing by looking at its graph?
  • To determine if a function is increasing by examining its graph, you should look at the slope of the curve. If as you move from left to right on the graph, the curve rises without any dips, then it indicates that the function is increasing. Additionally, you can check specific points: if higher input values correspond to higher output values across the domain shown in the graph, then it confirms that the function is indeed increasing.
• What effect do transformations have on the nature of a function being increasing or decreasing?
  • Transformations such as vertical shifts do not affect whether a function is increasing or decreasing. However, horizontal shifts and reflections can change intervals where a function increases or decreases. For example, reflecting a graph over the x-axis would convert an increasing function into a decreasing one. Understanding how these transformations interact with the characteristics of functions helps clarify their behavior when graphed.
• Evaluate how identifying increasing functions can assist in understanding real-world applications such as economics or biology.
  • Identifying increasing functions is crucial in real-world contexts like economics or biology because it helps predict trends and behaviors. For instance, in economics, an increasing function might represent rising prices over time or growing demand for a product. In biology, it could illustrate population growth rates. By recognizing which functions are increasing, we can make informed decisions based on expected future values, enhancing our understanding of dynamic systems and enabling better forecasting and planning.

Related terms

Decreasing Function: A decreasing function is one where, for any two points in its domain, if one point has a greater input value, the output value is less. This results in a downward slope when graphed.

Piecewise Function: A piecewise function is defined by different expressions based on the input value. It may include both increasing and decreasing intervals.

Monotonic Function: A monotonic function is one that is either entirely non-increasing or non-decreasing throughout its domain, ensuring a consistent direction in its outputs.