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

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 .

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)
.
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 determine the intercepts of the functions \(f(x) = x^3 + 1\) and \(g(x) = x + 1\), let's identify their x-intercepts and y-intercepts.

1. Finding the y-intercepts:

   • For \(f(x)\): \[ f(0) = 0^3 + 1 = 1 \quad \text{(y-intercept: } (0, 1) \text{)} \]

   • For \(g(x)\): \[ g(0) = 0 + 1 = 1 \quad \text{(y-intercept: } (0, 1) \text{)} \]

   Both functions have the same y-intercept at \((0, 1)\).

2. Finding the x-intercepts:

   • For \(f(x)\): Set \(f(x) = 0\): \[ x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1 \quad \text{(x-intercept: } (-1, 0) \text{)} \]

   • For \(g(x)\): Set \(g(x) = 0\): \[ x + 1 = 0 \implies x = -1 \quad \text{(x-intercept: } (-1, 0) \text{)} \]

   Both functions also have the same x-intercept at \((-1, 0)\).

Given this information, we can compare the intercepts:

• Both functions have the same y-intercept \((0, 1)\).
• Both functions have the same x-intercept \((-1, 0)\).

Based on the options provided, the correct statement is:

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