Question

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)=x^3+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).


The graphs intersect at (1,2).


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).

Answer 1

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

1. **Finding the y-intercept:**
- The y-intercept occurs where \( x = 0 \).
- For \( f(0) = 0^3 + 1 = 1 \) → y-intercept is \( (0, 1) \).
- For \( g(0) = 0 + 1 = 1 \) → y-intercept is \( (0, 1) \).

2. **Finding the x-intercept:**
- The x-intercept occurs where \( f(x) = 0 \) and \( g(x) = 0 \).
- For \( f(x) = 0 \): \( x^3 + 1 = 0 \) → \( x^3 = -1 \) → \( x = -1 \) → x-intercept is \( (-1, 0) \).
- For \( g(x) = 0 \): \( x + 1 = 0 \) → \( x = -1 \) → x-intercept is \( (-1, 0) \).

3. **Checking the intersection points:**
- To find where the graphs intersect, set \( f(x) = g(x) \):
\[ x^3 + 1 = x + 1 \]
\[ x^3 = x \]
\[ x(x^2 - 1) = 0 \]
This gives us \( x = 0 \) or \( x^2 = 1 \) (so \( x = 1 \) or \( x = -1 \)).
- Evaluating these:
- For \( x = -1 \): \( f(-1) = 0 \) and \( g(-1) = 0 \) → they intersect at \( (-1, 0) \).
- For \( x = 0 \): \( f(0) = 1 \) and \( g(0) = 1 \) → they intersect at \( (0, 1) \).
- For \( x = 1 \): \( f(1) = 2 \) and \( g(1) = 2 \) → they intersect at \( (1, 2) \).

Now evaluating the statements:
- **Statement 1**: "Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1)." – **False**.
- **Statement 2**: "The graphs intersect at (1, 2)." – **True**.
- **Statement 3**: "The x- and y-intercepts are different for each function." – **False** (they share x- and y-intercepts).
- **Statement 4**: "Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1)." – **True**, but only the x-intercept statement is correct since they have the same intercepts.

Thus, the true statements are that "The graphs intersect at (1,2)" and "Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1)".

The best response based on the question is:

**The graphs intersect at (1, 2).**