Example 1: a linear increasing function

• Show that f(x) = 2x + 1 is increasing on the set of real numbers.
• Solution Take a = 1 and b = 2. Then f(1) = 3 and f(2) = 5, and the second value is larger. For any a < b, f(b) − f(a) = 2(b − a) > 0 because b − a > 0. So f is increasing on all real numbers.

Example 2: a linear decreasing function

• Study the monotonicity of g(x) = 4 − 3x.
• Solution g(a) − g(b) = (4 − 3a) − (4 − 3b) = 3(b − a) > 0 for every a < b. So g(a) > g(b), and g is decreasing on all real numbers.

Example 3: a quadratic function

• Study the monotonicity of f(x) = x² − 4x + 1.
• Solution Here a = 1 > 0, so the parabola opens upward. The vertex is at x₀ = −(−4) / (2 × 1) = 2. Therefore f is decreasing on ]−∞, 2] and increasing on [2, ∞[.

Example 4: an absolute value function

• Study the monotonicity of h(x) = |x − 1|.
• Solution When x < 1, h(x) = 1 − x, which is decreasing. When x ≥ 1, h(x) = x − 1, which is increasing. So h is decreasing on ]−∞, 1] and increasing on [1, ∞[.

Example 5: a function defined by two rules

• Let k(x) = x² when x ≤ 0, and k(x) = 2x when x > 0. Study its monotonicity.
• Solution On ]−∞, 0] the rule x² is decreasing, because as x increases toward zero, x² decreases. On ]0, ∞[ the rule 2x is increasing. Since k(0) = 0, the function is decreasing on ]−∞, 0] and increasing on [0, ∞[.

Example 6: finding a constant that gives a required monotonicity