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)=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 .
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).
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 .
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).
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To analyze the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), we will first identify the intercepts for each function from the provided table.
### Y-Intercepts
- **For \( f(x) \)**: The y-intercept occurs when \( x = 0 \). From the table, \( f(0) = 1 \), so the y-intercept is \( (0, 1) \).
- **For \( g(x) \)**: Similarly, \( g(0) = 1 \), so the y-intercept is also \( (0, 1) \).
### X-Intercepts
- **For \( f(x) \)**: The x-intercept occurs when \( f(x) = 0 \). From the table, \( f(-1) = 0 \), so the x-intercept is \( (-1, 0) \).
- **For \( g(x) \)**: The x-intercept occurs when \( g(x) = 0 \). From the table, \( g(-1) = 0 \), so the x-intercept is also \( (-1, 0) \).
### Conclusion
Both functions have:
- A y-intercept of \( (0, 1) \)
- An x-intercept of \( (-1, 0) \)
Now let's evaluate the statements provided in the question:
1. **The graphs intersect at (1, 2)** - This is incorrect; the intersection is not at this point.
2. **The x- and y-intercepts are different for each function** - This is incorrect; both functions share the same intercepts.
3. **Both functions have a y-intercept of (−1, 0) and an x-intercept of (0, 1)** - This is incorrect; the y-intercept is \( (0, 1) \), not \( (-1, 0) \).
4. **Both functions have an x-intercept of (−1, 0) and a y-intercept of (0, 1)** - This statement is **true**.
Thus, the correct answer is:
**Both functions have an x-intercept of (−1, 0) and a y-intercept of (0, 1).**
### Y-Intercepts
- **For \( f(x) \)**: The y-intercept occurs when \( x = 0 \). From the table, \( f(0) = 1 \), so the y-intercept is \( (0, 1) \).
- **For \( g(x) \)**: Similarly, \( g(0) = 1 \), so the y-intercept is also \( (0, 1) \).
### X-Intercepts
- **For \( f(x) \)**: The x-intercept occurs when \( f(x) = 0 \). From the table, \( f(-1) = 0 \), so the x-intercept is \( (-1, 0) \).
- **For \( g(x) \)**: The x-intercept occurs when \( g(x) = 0 \). From the table, \( g(-1) = 0 \), so the x-intercept is also \( (-1, 0) \).
### Conclusion
Both functions have:
- A y-intercept of \( (0, 1) \)
- An x-intercept of \( (-1, 0) \)
Now let's evaluate the statements provided in the question:
1. **The graphs intersect at (1, 2)** - This is incorrect; the intersection is not at this point.
2. **The x- and y-intercepts are different for each function** - This is incorrect; both functions share the same intercepts.
3. **Both functions have a y-intercept of (−1, 0) and an x-intercept of (0, 1)** - This is incorrect; the y-intercept is \( (0, 1) \), not \( (-1, 0) \).
4. **Both functions have an x-intercept of (−1, 0) and a y-intercept of (0, 1)** - This statement is **true**.
Thus, the correct answer is:
**Both functions have an x-intercept of (−1, 0) and a y-intercept of (0, 1).**
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