- Learn how to determine the intervals where a function is increasing
- Learn how to determine the intervals where a function is decreasing
- Learn how to determine the intervals where a function is constant

## How to Determine the Intervals Where a Function is Increasing, Decreasing, or Constant

A function f is increasing on any interval if, for any (x

_{1}) and (x

_{2}), we have: $$x_1 < x_2$$ $$f(x_1) < f(x_2)$$ This tells us as we move right from x

_{1}to x

_{2}, the y-values are getting larger. A function f is decreasing on any interval if, for any (x

_{1}) and (x

_{2}), we have: $$x_1 < x_2$$ $$f(x_1) > f(x_2)$$ This tells us as we move right from x

_{1}to x

_{2}, the y-values are getting smaller. A function f is constant on any interval if, for every (x

_{1}) and (x

_{2}), we have: $$f(x_1)=f(x_2)$$ This tells us as we move right from x

_{1}to x

_{2}, the y-values are constant. When we get the question of whether a function is increasing, decreasing, or constant on an interval, think about what happens to the y-values as the x-values go from left to right. Is the graph climbing (increasing), falling (decreasing), or flat (constant).

### Increasing, Decreasing, and Constant Functions

- A function is increasing on an open interval, if f(x
_{1}) < f(x_{2}) whenever x_{1}< x_{2}for any x_{1}and x_{2}in the interval - A function is decreasing on an open interval, if f(x
_{1}) > f(x_{2}) whenever x_{1}< x_{2}for any x_{1}and x_{2}in the interval - A function is constant on an open interval, if f(x
_{1}) = f(x_{2}) for any for any x_{1}and x_{2}in the interval

- Some textbooks use closed intervals when discussing this topic.
- A good article for reference: Math Doctors Article

- Additionally, some resources may use the terms strictly increasing and strictly decreasing.
- A good article for reference: Math is Fun Article

Example #1: Find the intervals on which f is increasing and on which f is decreasing. $$f(x)=x^3 - 3x^2$$ From the graph we see that f is increasing on the intervals (-∞, 0) and (2, ∞) and is decreasing on (0, 2). These intervals are always given in terms of the x-values. A common mistake is to try to give the intervals in terms of the y-values. If you are having trouble with this concept, drawing in a person walking may help. Example #2: Find the intervals on which f is increasing and on which f is decreasing. $$f(x)=x^4 - 2x^2$$ From the graph we see that f is increasing on the intervals (-1, 0) and (1, ∞) and is decreasing on (-∞, -1) and (0, 1). These intervals are always given in terms of the x-values. A common mistake is to try to give the intervals in terms of the y-values. If you are having trouble with this concept, drawing in a person walking may help.

#### Skills Check:

Example #1

Determine if the following is true or false. $$f(x)=x^2 - 3x$$ Decreasing on: $$\left(-\infty, \frac{3}{2}\right)$$ Increasing on: $$\left(\frac{3}{2}, \infty\right)$$

Please choose the best answer.

Example #2

Determine if the following is true or false. $$f(x)=\large{x}^{\frac{2}{3}}- 1$$ Decreasing on: $$\left(-\infty, -1\right)$$ Increasing on: $$\left(-1, \infty\right)$$

Please choose the best answer.

Example #3

Sketch the graph and find the interval on which the function is increasing. $$\ f(x)=\begin{cases}-2x + 3 & \text{if}\hspace{.2em}x ≤ 5 \\ 3x + 1 & \text{if}\hspace{.2em}x > 5 \end{cases}$$

Please choose the best answer.

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