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
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.
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).
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.
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).
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine the correct statement about the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), let's first identify the x- and y-intercepts of each function.
**Finding the intercepts:**
1. **Y-Intercepts:**
- For \( f(x) \): Evaluate \( f(0) = 0^3 + 1 = 1 \). So, the y-intercept is \( (0, 1) \).
- For \( g(x) \): Evaluate \( g(0) = 0 + 1 = 1 \). So, the y-intercept is also \( (0, 1) \).
2. **X-Intercepts:**
- For \( f(x) \): Set \( f(x) = 0 \):
\[
x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1.
\]
So, the x-intercept is \( (-1, 0) \).
- For \( g(x) \): Set \( g(x) = 0 \):
\[
x + 1 = 0 \implies x = -1.
\]
So, the x-intercept is also \( (-1, 0) \).
**Comparison of Intercepts:**
- Both functions have the same y-intercept of \( (0, 1) \).
- Both functions have the same x-intercept of \( (-1, 0) \).
Now, let's analyze the provided statements:
1. **"Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1)."**
- False.
2. **"Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1)."**
- False.
3. **"The x- and y-intercepts are different for each function."**
- False.
4. **"Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1)."**
- True.
5. **"The graphs intersect at (1,2)."**
- False because at \( x = 1 \); \( f(1) = 2 \) and \( g(1) = 2 \) indicate they intersect at \( (1, 2) \), but this statement is focused on intercepts.
Therefore, 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)."**
**Finding the intercepts:**
1. **Y-Intercepts:**
- For \( f(x) \): Evaluate \( f(0) = 0^3 + 1 = 1 \). So, the y-intercept is \( (0, 1) \).
- For \( g(x) \): Evaluate \( g(0) = 0 + 1 = 1 \). So, the y-intercept is also \( (0, 1) \).
2. **X-Intercepts:**
- For \( f(x) \): Set \( f(x) = 0 \):
\[
x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1.
\]
So, the x-intercept is \( (-1, 0) \).
- For \( g(x) \): Set \( g(x) = 0 \):
\[
x + 1 = 0 \implies x = -1.
\]
So, the x-intercept is also \( (-1, 0) \).
**Comparison of Intercepts:**
- Both functions have the same y-intercept of \( (0, 1) \).
- Both functions have the same x-intercept of \( (-1, 0) \).
Now, let's analyze the provided statements:
1. **"Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1)."**
- False.
2. **"Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1)."**
- False.
3. **"The x- and y-intercepts are different for each function."**
- False.
4. **"Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1)."**
- True.
5. **"The graphs intersect at (1,2)."**
- False because at \( x = 1 \); \( f(1) = 2 \) and \( g(1) = 2 \) indicate they intersect at \( (1, 2) \), but this statement is focused on intercepts.
Therefore, 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)."**
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