Question

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 x- and y-intercepts are different for each function. The x - and y -intercepts are different for each function. Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1). Both functions have an x -intercept of left parenthesis negative 1 comma 0 right parenthesis and a y -intercept of left parenthesis 0 comma 1 right parenthesis . 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 . The graphs intersect at (1,2).

Answer 1

To compare the intercepts of the functions \( f(x)=x^3+1 \) and \( g(x)=x+1 \), we will need to find their x-intercepts and y-intercepts.

### For \( f(x)=x^3+1 \)

1. **Y-intercept**: This occurs when \( x = 0 \).
\[
f(0) = 0^3 + 1 = 1
\]
Thus, the y-intercept is \( (0, 1) \).

2. **X-intercept**: This occurs when \( f(x) = 0 \).
\[
x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1
\]
Thus, the x-intercept is \( (-1, 0) \).

### For \( g(x)=x+1 \)

1. **Y-intercept**: This occurs when \( x = 0 \).
\[
g(0) = 0 + 1 = 1
\]
Thus, the y-intercept is \( (0, 1) \).

2. **X-intercept**: This occurs when \( g(x) = 0 \).
\[
x + 1 = 0 \implies x = -1
\]
Thus, the x-intercept is \( (-1, 0) \).

### Results Summary

- **Y-intercepts**: Both functions have a y-intercept of \( (0, 1) \).
- **X-intercepts**: Both functions have an x-intercept of \( (-1, 0) \).

### Conclusion

Based on the intercepts determined:

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

Therefore, the correct statement is:

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