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.

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 .

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

Answer 1

Let's analyze the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \).

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

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

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

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

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

Given this analysis, let's evaluate the provided statements:

- **The x- and y-intercepts are different for each function.** (False)
- **Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1).** (False)
- **Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1).** (True)
- **The graphs intersect at (1,2).** (False)

Therefore, the true statement is:

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