Question

The solid curve is labeled f of x and passes through left parenthesis 3 comma 2.7 right parenthesis, left parenthesis 2 comma 2.4 right parenthesis, and left parenthesis 0 comma 1 right parenthesis. The dotted curve is labeled g of x and passes through left parenthesis 3 comma 2.5 right parenthesis, left parenthesis 1 comma 2 right parenthesis, left parenthesis 0 comma 1 right parenthesis, left parenthesis negative 1 comma 0 right parenthesis, and left parenthesis negative 3 comma 0.3 right parenthesis.

Compare the graphs of f(x)=x−−√+1
and g(x)=x−−√3+1
. Which one of the following options is true?

(1 point)

Both functions are increasing on their domain.

Both functions go through the points (−1,0)
and (0,1)
.

Both functions have the same domain, x>0
.

Both functions have the same range, (−∞,∞)
.
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Answer 1

"Both functions are increasing on their domain."

Reason: sqrt(x)+1 is increasing for x≥0 (its derivative 1/(2√x) > 0 for x>0 and the function is nondecreasing at 0), and the cube-root function ∛x+1 is increasing for all real x (∛x is strictly increasing). The other choices are false (√x+1 is not defined at x = −1, domains and ranges differ).