Use the table to answer the question.

Question

x	 f(x)=x3+1 	 g(x)=x+1 
−2 	 −7 	 −1 
−1 	0	0
0	1	1
1	2	2
2	9	3
Compare the functions  f(x)=x3+1  and  g(x)=x+1 . Which of the following statements is true about the intercepts of the functions?

(1 point)
Responses

The graphs intersect at (1,2).
The graphs intersect at  left parenthesis 1 comma 2 right parenthesis .

The x- and y-intercepts are different for each function.
The x - and y -intercepts are different for each function.

Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1).
Both functions have a y -intercept of  left parenthesis negative 1 comma 0 right parenthesis  and an x -intercept of  left parenthesis 0 comma 1 right parenthesis .

Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1).

GPT-4o mini · Answer

To analyze the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), we will first identify the intercepts for each function from the provided table.

Y-Intercepts

For \( f(x) \): The y-intercept occurs when \( x = 0 \). From the table, \( f(0) = 1 \), so the y-intercept is \( (0, 1) \).
For \( g(x) \): Similarly, \( g(0) = 1 \), so the y-intercept is also \( (0, 1) \).

X-Intercepts

For \( f(x) \): The x-intercept occurs when \( f(x) = 0 \). From the table, \( f(-1) = 0 \), so the x-intercept is \( (-1, 0) \).
For \( g(x) \): The x-intercept occurs when \( g(x) = 0 \). From the table, \( g(-1) = 0 \), so the x-intercept is also \( (-1, 0) \).

Conclusion

Both functions have:

A y-intercept of \( (0, 1) \)
An x-intercept of \( (-1, 0) \)

Now let's evaluate the statements provided in the question:

The graphs intersect at (1, 2) - This is incorrect; the intersection is not at this point.
The x- and y-intercepts are different for each function - This is incorrect; both functions share the same intercepts.
Both functions have a y-intercept of (−1, 0) and an x-intercept of (0, 1) - This is incorrect; the y-intercept is \( (0, 1) \), not \( (-1, 0) \).
Both functions have an x-intercept of (−1, 0) and a y-intercept of (0, 1) - This statement is true.

Thus, the correct answer is: Both functions have an x-intercept of (−1, 0) and a y-intercept of (0, 1).