Asked by sh

The solid curve is labeled as f left parenthesis x right parenthesis equals Start Root x plus 1 End Root minus 2. The solid curve starts in the third quadrant from left parenthesis negative 1 comma negative 2 right parenthesis. The solid curve increases through left parenthesis 0 comma negative 1 right parenthesis and through the fourth quadrant. It ends with an arrow approximately at left parenthesis 2.3 comma negative 0.2 right parenthesis. The dotted curve is labeled as g left parenthesis x right parenthesis equals root index 3 Start Root x plus 1 End Root minus 2. The dotted curve has arrows at both its ends. The dotted curve starts in the third quadrant at approximately left parenthesis negative 3 comma negative 3.2 right parenthesis. The dotted curve increases through the third quadrant towards left parenthesis negative 1 comma negative 2 right parenthesis, where it briefly, vertically increases in parallel with the y-axis before increasing past left parenthesis 0 comma negative 1 right parenthesis towards the fourth quadrant. The dotted curve approximately ends at left parenthesis 2.3 comma negative 0.5 right parenthesis.


Compare the graphs of f(x)=x+1−−−−√−2 and g(x)=x+1−−−−√3−2 . Identify which of the following features is true about the graphs.

(1 point)
Responses

The graphs have the same y-intercept.
The graphs have the same y -intercept.

The graphs have the same domains.
The graphs have the same domains.

The graphs have the same ranges.
The graphs have the same ranges.

The graphs have the same x-intercept.
The graphs have the same x -intercept.

All Answers 1

Answered by GPT-5 mini AI
Only the first statement is true: the graphs have the same y‑intercept.

Reasons:
- y‑intercept: f(0)=√(0+1)−2=1−2=−1 and g(0)=∛(0+1)−2=1−2=−1, so both pass through (0,−1).
- domain: f requires x+1≥0 → domain f = [−1,∞); g is defined for all real x → domain g = (−∞,∞). Not the same.
- range: f(x)≥−2 → range f = [−2,∞); g takes all real values → range g = (−∞,∞). Not the same.
- x‑intercept: f: √(x+1)−2=0 → x=3; g: ∛(x+1)−2=0 → x=7. Not the same.