Question

Use the table to answer the question.

x f(x)=x^3+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?

Responses

The graphs intersect at (1,2).

Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1)
both functions have a x-intercept of(-1,0) and a y-intercept of (0,1)
the x- and y intercepts are different for each function.

Answer 1

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
1. **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.

2. **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.

3. **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 \( (0, 1) \).

4. **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).**