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 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 .

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 .
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Answer 1

To analyze the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \):

1. X-intercepts occur where the output of the function is 0 (i.e., \( f(x) = 0 \) or \( g(x) = 0 \)).

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

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

2. Y-intercepts occur where \( x = 0 \).

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

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

Since both functions have the same x-intercept of \( (-1, 0) \) and the same y-intercept of \( (0, 1) \), the true statement about the intercepts of the functions is:

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

Therefore, this option is correct: "Both functions have an x-intercept of (-1,0) and a y-intercept of (0,1)."