Increasing, Decreasing, Max, and Min Functions

Increasing and Decreasing Functions

  • An increasing function is one where the y values go up as the x values increase from left to right.
    • Formally, if x2 > x1, then f(x2) > f(x1).
  • A decreasing function is one where the y values go down as the x values increase from left to right.
    • Formally, if x2 > x1, then f(x2) < f(x1).

Identifying Intervals of Increase and Decrease

  • To describe where a function is increasing or decreasing, use intervals based on the x values.
  • Example:
    • Decreasing: For x values between -4 and -2, and between 4 and 7. Represented as -4 < x < -2 and 4 < x < 7.
    • Increasing: For x values between -2 and 1. Represented as -2 < x < 1.
  • Interval notation:
    • Decreasing: [-4, -2) \cup (4, 7)
    • Increasing: (-2, 1)
  • If the graph has arrows indicating it continues indefinitely, the intervals extend to infinity.
    • Decreasing: (-\infty, -2) \cup (4, \infty)
    • Increasing: (-2, 1)

Key Points

  • When considering increasing and decreasing parts of functions from graphs, remember that x increases from left to right.
  • Increasing: y goes up.
  • Decreasing: y goes down.
  • Always describe intervals in terms of x values.

Maximums and Minimums of Functions

Absolute Maximum

  • A function f(x) has an absolute maximum at x = c if f(c) is the largest y value in the entire domain of f.
    • f(c) \ge f(x) for all x in the domain of f.
  • f(c) is the absolute maximum value.
  • The point (c, f(c)) is the absolute maximum point.
  • A function can have multiple absolute maximum points if the highest y value is achieved at multiple x values, but it can only have one absolute maximum value.

Absolute Minimum

  • A function f(x) has an absolute minimum at x = c if f(c) is the smallest y value in the entire domain of f.
    • f(c) \le f(x) for all x in the domain of f.
  • f(c) is the absolute minimum value.
  • The point (c, f(c)) is the absolute minimum point.

Global Maximum and Minimum

  • Absolute maximum and minimum values are also called global maximum and minimum values.

Local Maximum

  • A function f(x) has a local maximum at x = c if f(c) is larger than any y value nearby.
    • f(c) \ge f(x) for all x in an open interval around c.
  • f(c) is the local maximum value.
  • (c, f(c)) is the local maximum point.

Local Minimum

  • A function f(x) has a local minimum at x = c if f(c) is smaller than or equal to f(x) for all x values in an open interval around c.
  • f(c) is called a local minimum value.
  • (c, f(c)) is called a local minimum point.

Relative Maximum and Minimum

  • Maximum and minimum values can also be called relative maximum and minimum values.

Discontinuities

  • If an endpoint of a function's domain is the highest point nearby, some sources may consider it a local maximum, while others may not because an open interval around that point cannot be defined.