To determine the correct statements about the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), let's first calculate the intercepts for both functions.
1. Y-Intercepts
-
For \( f(x) = x^3 + 1 \):
- \( f(0) = 0^3 + 1 = 1 \)
- So, the y-intercept is \( (0, 1) \).
-
For \( g(x) = x + 1 \):
- \( g(0) = 0 + 1 = 1 \)
- So, the y-intercept is also \( (0, 1) \).
2. X-Intercepts
-
For \( f(x) = x^3 + 1 \):
- Set \( f(x) = 0 \): \[ x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1 \]
- So, the x-intercept is \( (-1, 0) \).
-
For \( g(x) = x + 1 \):
- Set \( g(x) = 0 \): \[ x + 1 = 0 \implies x = -1 \]
- So, the x-intercept is also \( (-1, 0) \).
Summary of Intercepts
- Both functions have a y-intercept of \( (0, 1) \).
- Both functions have an x-intercept of \( (-1, 0) \).
Analyzing the Given Statements
-
The graphs intersect at (1,2).
- This is false; at \( x = 1 \), \( f(1) = 2 \) and \( g(1) = 2 \), so they do intersect at that point, but it's not about the intercepts.
-
Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1).
- This is false; they both have a y-intercept of \( (0, 1) \) and an x-intercept of \( (-1, 0) \), not the other way around.
-
Both functions have a x-intercept of(-1,0) and a y-intercept of (0,1).
- This is true; both functions have an x-intercept of \( (-1, 0) \) and a y-intercept of \( (0, 1) \).
-
The x- and y-intercepts are different for each function.
- This is false; the x- and y-intercepts are the same for both functions.
Conclusion
The correct statement is: Both functions have an x-intercept of (-1,0) and a y-intercept of (0,1).