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

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 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 an x-intercept of (−1,0)
and a y-intercept of (0,1)
.

Answer 1

To compare the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \) and to analyze their intercepts, let’s calculate their x-intercepts and y-intercepts.

### Finding the intercepts

1. **y-intercept**: This is found by evaluating the function at \( x = 0 \).

- For \( f(x) \):
\[
f(0) = 0^3 + 1 = 1
\]
So the y-intercept of \( f(x) \) is \( (0, 1) \).

- For \( g(x) \):
\[
g(0) = 0 + 1 = 1
\]
So the y-intercept of \( g(x) \) is also \( (0, 1) \).

2. **x-intercept**: This is found by evaluating the function when \( f(x) = 0 \) or \( g(x) = 0 \).

- For \( f(x) \):
\[
x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1
\]
So the x-intercept of \( f(x) \) is \( (-1, 0) \).

- For \( g(x) \):
\[
x + 1 = 0 \implies x = -1
\]
So the x-intercept of \( g(x) \) is \( (-1, 0) \).

### Summary of intercepts

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

### Conclusion

The statement that both functions have an x-intercept of \( (-1, 0) \) and a y-intercept of \( (0, 1) \) is true. Therefore, the correct response is:

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