Given the following function: log₂ (x+5) - 1

Question

Determine the characteristics for this function:

Domain:

Range:

X-intercept:

Asymptote:

Increasing Interval:

Decreasing Interval:

Left End Behavior:

Right End Behavior:

Answer 1

This is a log (base 2) function shifted 5 units left and 1 unit down. The shift down won't affect anything except the location of the x-intercept.

Domain: x > - 5 (since we can only take the log of + numbers)

Range: All real numbers (this is the range of an untransformed log function and it does not change)

x-intercept: log₂(x+5) - 1 = 0. log₂(x+5) = 1 (x+5) = 2^1. x = - 3.

Vertical asymptote: x = -5 (shifted 5 units left from the y-axis, which is the VA of an untransformed log function)

Increasing: All real numbers (This is the behavior of any log function with base > 1.)

Decreasing: nowhere

Left end behavior: As x → -5⁺ , y decreases without bound (y → - ∞)

Right end behavior: As x → ∞ , y increases without bound (y → ∞)