Compare the graphs of f (x) = √x +1 and g(x) = ∛x +1. Which one of the following options is true?
• Both functions go through the points (-1, 0) and (0, 1).
• Both functions are increasing on their domain.
• Both functions have the same range, (-∞, co).
• Both functions have the same domain, x > 0.
The correct option is: Both functions have the same domain, x > 0.
For both functions f(x) = √x + 1 and g(x) = ∛x + 1,
1. Both functions go through the points (-1, 0) and (0, 1):
For f(x): f(-1) = √(-1) + 1 = not defined
f(0) = √0 + 1 = 1
For g(x): g(-1) = ∛(-1) + 1 = 0
g(0) = ∛0 + 1 = 1
Hence, the given statement is not true.
2. Both functions are increasing on their domain:
For f(x): As x increases, f(x) also increases. Hence, it is increasing on its domain.
For g(x): As x increases, g(x) also increases. Hence, it is increasing on its domain.
Hence, the given statement is true.
3. Both functions have the same range, (-∞, ∞):
For f(x): As x increases, f(x) also increases. Hence, f(x) can take any positive value. Therefore, the range of f(x) is (0, ∞).
For g(x): As x increases, g(x) also increases. Hence, g(x) can take any positive value. Therefore, the range of g(x) is (0, ∞).
Hence, the given statement is true.
4. Both functions have the same domain, x > 0:
For f(x): No restriction on x except that x should be greater than or equal to 0. Therefore, the domain of f(x) is [0, ∞).
For g(x): No restriction on x except that x should be greater than or equal to 0. Therefore, the domain of g(x) is [0, ∞).
Hence, the given statement is true.
Thus, the correct option is: Both functions have the same domain, x > 0.