Monotonicity Math Example 3
Follow the full solution, then compare it with the other examples linked below.
Example 3
mediumFor \(f(x) = -x + 5\), is it increasing or decreasing? Verify with two test values.
Solution
- 1 The slope is \(-1 < 0\), so the function is decreasing.
- 2 Test: \(f(1) = 4\), \(f(3) = 2\). Since \(1 < 3\) and \(4 > 2\): larger input → smaller output.
- 3 \(f\) is monotonically decreasing.
Answer
Monotonically decreasing
Negative slope means the function decreases as \(x\) increases. Confirmed: \(f(1)=4 > f(3)=2\).
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 1 medium
Is (f(x) = 2x + 3) monotonically increasing? Show that if (x_1 < x_2) then (f(x_1) < f(x_2)).
Example 2 hardDetermine the intervals on which (h(x) = x^3 - 3x) is increasing and decreasing.
Example 4 hardShow that (f(x) = x^2) is NOT monotone on all of (mathbb{R}) by giving a counterexample, then state