Question

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)

The graphs have the same x-intercept.

The graphs have the same y-intercept.

The graphs have the same domains.

The graphs have the same ranges.
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Answer 1

Only "The graphs have the same y-intercept" is true.

Reasoning:
- y-intercepts: f(0)=√(0+1)−2=1−2=−1 and g(0)=∛(0+1)−2=1−2=−1, so both have (0,−1).
- x-intercepts: f(x)=0 ⇒ √(x+1)=2 ⇒ x=3; g(x)=0 ⇒ ∛(x+1)=2 ⇒ x=7, so different.
- domains: f: x≥−1; g: all real x, so different.
- ranges: f: y≥−2; g: all real y, so different.