Question

Use the table to answer the 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 x- and y-intercepts are different for each function.
The x - and y -intercepts are different for each function.

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

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 .
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Answer 1

To analyze the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), we first need to determine the x-intercepts (where \( f(x) = 0 \) or \( g(x) = 0 \)) and the y-intercepts (where the functions are evaluated at \( x = 0 \)).

### Finding the y-intercepts:
1. **For \( f(x) = x^3 + 1 \):**
- \( f(0) = 0^3 + 1 = 1 \)
- Y-intercept: \( (0, 1) \)

2. **For \( g(x) = x + 1 \):**
- \( g(0) = 0 + 1 = 1 \)
- Y-intercept: \( (0, 1) \)

### Finding the x-intercepts:
1. **For \( f(x) = x^3 + 1 \):**
- Set \( f(x) = 0 \):
\( x^3 + 1 = 0 \)
\( x^3 = -1 \)
\( x = -1 \)
- X-intercept: \( (-1, 0) \)

2. **For \( g(x) = x + 1 \):**
- Set \( g(x) = 0 \):
\( x + 1 = 0 \)
\( x = -1 \)
- X-intercept: \( (-1, 0) \)

### Summary of Intercepts:
- Both functions have the same y-intercept: \( (0, 1) \).
- Both functions have the same x-intercept: \( (-1, 0) \).

### Comparison of Statements:
1. **The x- and y-intercepts are different for each function.**
- False

2. **The graphs intersect at \( (1,2) \).**
- True; at \( x = 1 \), \( f(1) = 2 \) and \( g(1) = 2 \), they intersect at \( (1, 2) \).

3. **Both functions have a y-intercept of \( (-1,0) \) and an x-intercept of \( (0,1) \).**
- False (intercepts are incorrectly stated).

4. **Both functions have an x-intercept of \( (-1,0) \) and a y-intercept of \( (0,1) \).**
- True (this is precisely correct).

From the options, the true statements are:
- The graphs intersect at \( (1, 2) \).
- Both functions have an x-intercept of \( (-1,0) \) and a y-intercept of \( (0,1) \).

Since the prompt may only want to select one option, the more direct answers are the statements regarding the intercepts:

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