Monotonicity Math Example 1
Follow the full solution, then compare it with the other examples linked below.
Example 1
mediumIs \(f(x) = 2x + 3\) monotonically increasing? Show that if \(x_1 < x_2\) then \(f(x_1) < f(x_2)\).
Solution
- 1 Assume \(x_1 < x_2\).
- 2 Multiply by 2 (positive, preserves inequality): \(2x_1 < 2x_2\).
- 3 Add 3 to both sides: \(2x_1 + 3 < 2x_2 + 3\).
- 4 So \(f(x_1) < f(x_2)\). ✓
- 5 \(f\) is monotonically increasing on all of \(\mathbb{R}\).
Answer
Yes — \(f(x) = 2x+3\) is strictly increasing
A function is monotonically increasing when larger inputs always produce larger outputs. Here the positive slope (2) guarantees this.
About Monotonicity
A function or sequence that consistently moves in one direction only—always increasing or always decreasing throughout its domain.
Learn more about Monotonicity →More Monotonicity Examples
Example 2 hard
Determine the intervals on which (h(x) = x^3 - 3x) is increasing and decreasing.
Example 3 mediumFor (f(x) = -x + 5), is it increasing or decreasing? Verify with two test values.
Example 4 hardShow that (f(x) = x^2) is NOT monotone on all of (mathbb{R}) by giving a counterexample, then state