Compare the graphs f(x) = sqrt(x + 1) - 2 and g(x) = ^3sqrt(x+ 1 - 2 Identify which of the following features is true about the graphs.
A The graphs have the same X intercept
B The graphs have the same Y intercepts
C The graphs have the same domains
D The graphs have the same ranges
To compare the graphs f(x) = √(x + 1) - 2 and g(x) = ∛(x + 1) - 2, let's analyze each feature:
A. X-intercepts are the points in which the graph intersects the x-axis. To find the x-intercept, we set y = 0 and solve for x.
For f(x):
0 = √(x + 1) - 2
2 = √(x + 1)
4 = x + 1
x = 3
So the x-intercept for f(x) is x = 3.
For g(x):
0 = ∛(x + 1) - 2
2 = ∛(x + 1)
8 = x + 1
x = 7
So the x-intercept for g(x) is x = 7.
Since the x-intercepts for f(x) and g(x) are different, option A is false.
B. Y-intercepts are the points in which the graph intersects the y-axis. To find the y-intercept, we set x = 0 and solve for y.
For f(x):
y = √(0 + 1) - 2
y = 1 - 2
y = -1
So the y-intercept for f(x) is y = -1.
For g(x):
y = ∛(0 + 1) - 2
y = 1 - 2
y = -1
So the y-intercept for g(x) is y = -1.
Since the y-intercepts for f(x) and g(x) are the same, option B is true.
C. The domain is the set of all possible x-values for which the function is defined.
For both f(x) and g(x), the domain includes all real numbers, since there are no restrictions or exclusions in the square root or cube root functions.
Thus, the graphs of f(x) = √(x + 1) - 2 and g(x) = ∛(x + 1) - 2 have the same domains. Option C is true.
D. The range is the set of all possible y-values that the function can take.
For f(x), since we have a square root function, the range will be all y-values greater than or equal to -2.
For g(x), since we have a cube root function, the range will be all y-values greater than or equal to -2.
Thus, the graphs of f(x) = √(x + 1) - 2 and g(x) = ∛(x + 1) - 2 have the same ranges. Option D is true.
In summary:
A. False
B. True
C. True
D. True
Only 1 answer
The correct answer is:
D. The graphs have the same ranges.