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 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 determine the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), we can find both the x-intercepts and y-intercepts for each function.

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

• Y-Intercept: Set \( x = 0 \): \[ f(0) = 0^3 + 1 = 1 \] So, the y-intercept is \( (0, 1) \).

• X-Intercept: Set \( f(x) = 0 \): \[ x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1 \] So, the x-intercept is \( (-1, 0) \).

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

• Y-Intercept: Set \( x = 0 \): \[ g(0) = 0 + 1 = 1 \] So, the y-intercept is \( (0, 1) \).

• X-Intercept: Set \( g(x) = 0 \): \[ x + 1 = 0 \implies x = -1 \] So, the x-intercept is \( (-1, 0) \).

Summary of Intercepts:

• For \( f(x) = x^3 + 1 \):

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

• For \( g(x) = x + 1 \):

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

Conclusion:

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

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