Question

The solid line is in quadrant 1 and starts at left parenthesis 1 comma 0 right parenthesis and curves up through left parenthesis 2 comma 1 right parenthesis. It continues upward to the left passing through left parenthesis 3 comma 1.5 right parenthesis, left parenthesis 4 comma 1.7 right parenthesis, and ends at left parenthesis 5 comma 2 right parenthesis with an arrow at the end. The equation of the line is in quadrant 1 above the solid line and is as follows: f left parenthesis x right parenthesis equals start root x minus 1 end root. The dotted line is an elongated S shape and has arrows at both ends. From left to right, it travels from quadrant 3 to quadrant 4, and up to quadrant 1. The left arrow is placed at left parenthesis negative 2 comma negative 1.8 right parenthesis. The line continues upward, toward the right, passing through left parenthesis negative 1 comma negative 1.5 right parenthesis, left parenthesis 0 comma negative 1.4 right parenthesis, left parenthesis 1 comma negative 1.5 right parenthesis, left parenthesis 0 comma negative 1.4 right parenthesis, left parenthesis 1 comma negative 1.2 right parenthesis, left parenthesis 2 comma negative 1 right parenthesis. The line then curves up to left parenthesis 3 comma 0 right parenthesis, travels straight up to left parenthesis 3 comma 0.5 right parenthesis, and curves upward to the right to left parenthesis 4 comma 1 right parenthesis, and ends with the arrow placed at left parenthesis 5 comma 1.3 right parenthesis. The equation of the curve is placed below the dotted line in the bottom, right quadrant and reads as follows: g left parenthesis x right parenthesis equals root index 3 start root x minus 3 end root.

Given the graphs of the functions and
, which one of the following statements is true?
(1 point)
f (x) = x − 1 −−−− √
g (x) = x − 3 −−−− √
3
The functions have the same domain.
The functions have the same range.
The values of for all values of x such that
.
f (x) > g (x)
x ≥ 1
The values of for all values of x such that
.
g (x) > f (x)
0 ≤ x < ∞

Answer 1

Correct choice: f(x) > g(x) for all x ≥ 1.

Reason:
- f(x) = sqrt(x−1) has domain x ≥ 1 and range [0, ∞). g(x) = ∛(x−3) has domain (−∞, ∞) and range (−∞, ∞). So they do not have the same domain or range.
- For 1 ≤ x < 3, g(x) = ∛(x−3) < 0 while f(x) ≥ 0, so f(x) > g(x).
- For x ≥ 3, set y = f(x) = √(x−1) (so y ≥ √2). Then g(x) = ∛(x−3) = ∛(y^2 − 2). Since cube is increasing, f(x) > g(x) is equivalent to
y > ∛(y^2 − 2) ⇔ y^3 > y^2 − 2 ⇔ y^3 − y^2 + 2 > 0.
For y ≥ √2 we have y^3 − y^2 + 2 ≥ 2√2 > 0, so the inequality holds.

Therefore f(x) > g(x) for every x ≥ 1.