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).
1 answer