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 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 .
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 .
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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 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 .
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 .
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page 12 of 12
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 \( g(x) = x + 1 \):
1. **X-intercepts** occur where the output of the function is 0 (i.e., \( f(x) = 0 \) or \( g(x) = 0 \)).
- For \( f(x) = x^3 + 1 \):
\[
x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1.
\]
Thus, the x-intercept is \( (-1, 0) \).
- For \( g(x) = x + 1 \):
\[
x + 1 = 0 \implies x = -1.
\]
Thus, the x-intercept is also \( (-1, 0) \).
2. **Y-intercepts** occur where \( x = 0 \).
- For \( f(x) = x^3 + 1 \):
\[
f(0) = 0^3 + 1 = 1.
\]
Thus, the y-intercept is \( (0, 1) \).
- For \( g(x) = x + 1 \):
\[
g(0) = 0 + 1 = 1.
\]
Thus, the y-intercept is also \( (0, 1) \).
Since both functions have the same x-intercept of \( (-1, 0) \) and the same y-intercept of \( (0, 1) \), the true statement about the intercepts of the functions is:
**Both functions have an x-intercept of \((-1, 0)\) and a y-intercept of \((0, 1)\).**
Therefore, this option is correct:
"Both functions have an x-intercept of (-1,0) and a y-intercept of (0,1)."
1. **X-intercepts** occur where the output of the function is 0 (i.e., \( f(x) = 0 \) or \( g(x) = 0 \)).
- For \( f(x) = x^3 + 1 \):
\[
x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1.
\]
Thus, the x-intercept is \( (-1, 0) \).
- For \( g(x) = x + 1 \):
\[
x + 1 = 0 \implies x = -1.
\]
Thus, the x-intercept is also \( (-1, 0) \).
2. **Y-intercepts** occur where \( x = 0 \).
- For \( f(x) = x^3 + 1 \):
\[
f(0) = 0^3 + 1 = 1.
\]
Thus, the y-intercept is \( (0, 1) \).
- For \( g(x) = x + 1 \):
\[
g(0) = 0 + 1 = 1.
\]
Thus, the y-intercept is also \( (0, 1) \).
Since both functions have the same x-intercept of \( (-1, 0) \) and the same y-intercept of \( (0, 1) \), the true statement about the intercepts of the functions is:
**Both functions have an x-intercept of \((-1, 0)\) and a y-intercept of \((0, 1)\).**
Therefore, this option is correct:
"Both functions have an x-intercept of (-1,0) and a y-intercept of (0,1)."
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