Calculus — Important Questions

f(x) = 3x² + 2x - 1, g(x) = x³ - 5x + 7, m(x) = 5x⁴ - 3x² + 8

Polynomial functions have terms with non-negative integer powers of x. f(x), g(x), and m(x) satisfy this condition. h(x) has x⁻¹ (negative power) and k(x) has x^(1/2) (fractional power), so they are not polynomials.

True

For continuity at x = 0: lim(x→0⁻) |x| = 0, lim(x→0⁺) |x| = 0, and f(0) = |0| = 0. Since all three values are equal, f(x) = |x| is continuous at x = 0.

f(x) = 2ˣ is always positive, f(x) = 3ˣ is strictly increasing, f(x) = (1/2)ˣ is strictly decreasing

For a > 1, aˣ is strictly increasing and always positive. For 0 < a < 1, aˣ is strictly decreasing and always positive. The domain of aˣ is all real numbers, not (0, ∞).