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

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 .

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 .

1 answer

Let's analyze the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \).

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

  1. Y-intercept: This occurs when \( x = 0 \): \[ f(0) = 0^3 + 1 = 1 \] So the y-intercept is \( (0, 1) \).

  2. X-intercept: This occurs when \( f(x) = 0 \): \[ x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1 \] So the x-intercept is \( (-1, 0) \).

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

  1. Y-intercept: This occurs when \( x = 0 \): \[ g(0) = 0 + 1 = 1 \] So the y-intercept is also \( (0, 1) \).

  2. X-intercept: This occurs when \( g(x) = 0 \): \[ x + 1 = 0 \implies x = -1 \] So the x-intercept is \( (-1, 0) \).

Summary of intercepts:

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

Given this analysis, let's evaluate the provided statements:

  • The x- and y-intercepts are different for each function. (False)
  • Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1). (False)
  • Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1). (True)
  • The graphs intersect at (1,2). (False)

Therefore, the true statement is:

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