Question

What is the domain, range, X-intercept, Y-intercept, relative extrema, end behavior, and the type of parent function of y=log2 (x+2).

Answer 1

- Parent function: logarithmic, y = log2(x) (log base 2).

- Domain: x + 2 > 0 ⇒ x ∈ (-2, ∞).

- Range: all real numbers, (-∞, ∞).

- Vertical asymptote: x = -2.

- X-intercept: set y = 0 ⇒ x + 2 = 1 ⇒ x = -1 ⇒ (−1, 0).

- Y-intercept: set x = 0 ⇒ y = log2(2) = 1 ⇒ (0, 1).

- Relative extrema: none (function is strictly increasing on its domain).

- End behavior:
- As x → −2+ (from the right), y → −∞.
- As x → ∞, y → ∞.

(Also: y′ = 1/((x+2) ln 2) > 0 so increasing; y″ = −1/((x+2)^2 ln 2) < 0 so the graph is concave down on its domain.)