Question

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 . 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) .

Answer 1

To analyze the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), we need to determine their x-intercepts and y-intercepts.

### Finding the x-intercepts:

1. **For \( f(x) = x^3 + 1 \):**
Set \( f(x) = 0 \):
\[
x^3 + 1 = 0 \Rightarrow x^3 = -1 \Rightarrow x = -1
\]
So, the x-intercept of \( f(x) \) is \( (-1, 0) \).

2. **For \( g(x) = x + 1 \):**
Set \( g(x) = 0 \):
\[
x + 1 = 0 \Rightarrow x = -1
\]
So, the x-intercept of \( g(x) \) is also \( (-1, 0) \).

### Finding the y-intercepts:

1. **For \( f(x) = x^3 + 1 \):**
Set \( x = 0 \):
\[
f(0) = 0^3 + 1 = 1
\]
So, the y-intercept of \( f(x) \) is \( (0, 1) \).

2. **For \( g(x) = x + 1 \):**
Set \( x = 0 \):
\[
g(0) = 0 + 1 = 1
\]
So, the y-intercept of \( g(x) \) is also \( (0, 1) \).

### Summary of Intercepts:
- Both functions have the same x-intercept: \( (-1, 0) \).
- Both functions have the same y-intercept: \( (0, 1) \).

### Final Conclusion:
The true statement about the intercepts of the functions is:
"Both functions have an x-intercept of \( (-1, 0) \) and a y-intercept of \( (0, 1) \)."