Use the table to answer the question.

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

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)
.

Answer 1

To determine the correct statements about the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), we first need to find the intercepts of both functions.

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

1. Y-Intercept: \[ f(0) = 0^3 + 1 = 1 \] The y-intercept is \( (0, 1) \).

2. X-Intercept: Set \( f(x) = 0 \): \[ x^3 + 1 = 0 \Rightarrow x^3 = -1 \Rightarrow x = -1 \] The x-intercept is \( (-1, 0) \).

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

1. Y-Intercept: \[ g(0) = 0 + 1 = 1 \] The y-intercept is \( (0, 1) \).

2. X-Intercept: Set \( g(x) = 0 \): \[ x + 1 = 0 \Rightarrow x = -1 \] The x-intercept is \( (-1, 0) \).

Summary of Intercepts:

• \( f(x) = x^3 + 1 \):

  • X-Intercept: \( (-1, 0) \)
  • Y-Intercept: \( (0, 1) \)

• \( g(x) = x + 1 \):

  • X-Intercept: \( (-1, 0) \)
  • Y-Intercept: \( (0, 1) \)

Analyzing the Statements:

• Both functions have a y-intercept of \( (0, 1) \) and an x-intercept of \( (-1, 0) \).
• The graphs intersect at \( (1, 2) \) does not apply because they do not share that point.
• The x- and y-intercepts are the same for both functions, specifically \( (-1, 0) \) and \( (0, 1) \).

The correct statement is: Both functions have an x-intercept of \( (-1, 0) \) and a y-intercept of \( (0, 1) \).