Compare the graphs of f(x)=x−−√+1 and g(x)=x−−√3+1. Which one of the following options is true?
A. Both functions are increasing on thier domain
B. Both functions have the same domain x>0
C. Both functions go through the points (-1,0) and (0,1)
D. Both functions have the same range
To compare the graphs, we can analyze the properties of the functions.
For f(x) = √(x) + 1, the domain is x ≥ 0 since the square root of a negative number is undefined in the real numbers.
For g(x) = √(3x) + 1, the domain is x ≥ 0 as well, since √(3x) is undefined for negative x.
Now let's analyze the properties of the functions:
1. Increasing/Decreasing:
For f(x) = √(x) + 1, the function is increasing on its domain. As x increases, √(x) also increases, and when we add 1, the function increases further.
For g(x) = √(3x) + 1, the function is also increasing on its domain. As x increases, √(3x) increases and adding 1 further increases the function.
Option A is correct: Both functions are increasing on their domain.
2. Domain:
As discussed above, both functions have the same domain: x ≥ 0.
Option B is correct: Both functions have the same domain x > 0.
3. Points on the graph:
For f(x) = √(x) + 1, plugging in x = -1 gives us f(-1) = √(-1) + 1, which is undefined since the square root of a negative number is undefined in the real numbers. So f(x) does not go through the point (-1, 0).
For g(x) = √(3x) + 1, plugging in x = -1 gives us g(-1) = √(3(-1)) + 1 = √(-3) + 1, which is again undefined since the square root of a negative number is undefined in the real numbers. So g(x) does not go through the point (-1, 0) either.
Option C is incorrect: Both functions do not go through the points (-1, 0) and (0, 1).
4. Range:
To find the range of f(x) = √(x) + 1, observe that √(x) has a range of [0, ∞), and adding 1 shifts the range to [1, ∞).
To find the range of g(x) = √(3x) + 1, observe that √(3x) has a range of [0, ∞), and adding 1 shifts the range to [1, ∞).
Option D is correct: Both functions have the same range.
In summary,
A. Both functions are increasing on their domain: True
B. Both functions have the same domain x > 0: True
C. Both functions go through the points (-1,0) and (0,1): False
D. Both functions have the same range: True