Question
Use the table to answer the question. x \(f\left(x\right)=x^3+1\) \(g\left(x\right)=x+1\) \(-2\) \(-7\) \(-1\) \(-1\) 0 0 0 1 1 1 2 2 2 9 3 Compare the functions \(f\left(x\right)=x^3+1\) and \(g\left(x\right)=x+1\). Which of the following statements is true about the intercepts of the functions? (1 point) Responses Both functions have an x-intercept of \(\left(-1,0\right)\) and a y-intercept of \(\left(0,1\right)\). 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 \(\left(1,2\right)\). 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 \(\left(-1,0\right)\) and an x-intercept of \(\left(0,1\right)\)
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Answer
can u pick the best one?
Answered by
GPT-4o mini
To determine the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), we need to find both the x-intercepts and y-intercepts for each function.
1. **Finding the y-intercepts:**
- For \( f(x) = x^3 + 1 \):
- The y-intercept occurs when \( x = 0 \):
\[
f(0) = 0^3 + 1 = 1 \quad \Rightarrow \text{y-intercept: } (0, 1)
\]
- For \( g(x) = x + 1 \):
- The y-intercept occurs when \( x = 0 \):
\[
g(0) = 0 + 1 = 1 \quad \Rightarrow \text{y-intercept: } (0, 1)
\]
2. **Finding the x-intercepts:**
- For \( f(x) = x^3 + 1 \):
- The x-intercept occurs when \( f(x) = 0 \):
\[
x^3 + 1 = 0 \quad \Rightarrow \quad x^3 = -1 \quad \Rightarrow \quad x = -1 \quad \Rightarrow \text{x-intercept: } (-1, 0)
\]
- For \( g(x) = x + 1 \):
- The x-intercept occurs when \( g(x) = 0 \):
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1 \quad \Rightarrow \text{x-intercept: } (-1, 0)
\]
3. **Finding points of intersection:**
To see if the graphs intersect at \( (1, 2) \):
- Calculate \( f(1) \) and \( g(1) \):
- \( f(1) = 1^3 + 1 = 2 \)
- \( g(1) = 1 + 1 = 2 \)
So, both functions do intersect at \( (1, 2) \).
Now, let's summarize the intercepts:
- Both functions have:
- X-intercept: \((-1, 0)\)
- Y-intercept: \((0, 1)\)
The graphs intersect at \((1, 2)\).
Now we can analyze the given responses:
- "Both functions have an x-intercept of \((-1, 0)\) and a y-intercept of \((0, 1)\)." (True)
- "The graphs intersect at \((1, 2)\)." (True)
- "The x- and y-intercepts are different for each function." (False)
- "Both functions have a y-intercept of \((-1, 0)\) and an x-intercept of \((0, 1)\)" (False)
Since the first two statements are both true based on the analysis, and depending on the context of the question where only one statement should be chosen, you may choose the one that emphasizes the intercepts specifically. Both functions have the same intercepts, and they intersect at \((1, 2)\).
Generally, the statement "Both functions have an x-intercept of \((-1, 0)\) and a y-intercept of \((0, 1)\)." would be the most comprehensive one, but since the statement "The graphs intersect at \((1, 2)\)." is also true and focuses on their behavior, it may also work depending on the question's intent.
If you need to choose one, it could be either:
1. "Both functions have an x-intercept of \((-1, 0)\) and a y-intercept of \((0, 1)\)."
2. "The graphs intersect at \((1, 2)\)."
Either can be deemed correct depending on what the question focuses on.
1. **Finding the y-intercepts:**
- For \( f(x) = x^3 + 1 \):
- The y-intercept occurs when \( x = 0 \):
\[
f(0) = 0^3 + 1 = 1 \quad \Rightarrow \text{y-intercept: } (0, 1)
\]
- For \( g(x) = x + 1 \):
- The y-intercept occurs when \( x = 0 \):
\[
g(0) = 0 + 1 = 1 \quad \Rightarrow \text{y-intercept: } (0, 1)
\]
2. **Finding the x-intercepts:**
- For \( f(x) = x^3 + 1 \):
- The x-intercept occurs when \( f(x) = 0 \):
\[
x^3 + 1 = 0 \quad \Rightarrow \quad x^3 = -1 \quad \Rightarrow \quad x = -1 \quad \Rightarrow \text{x-intercept: } (-1, 0)
\]
- For \( g(x) = x + 1 \):
- The x-intercept occurs when \( g(x) = 0 \):
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1 \quad \Rightarrow \text{x-intercept: } (-1, 0)
\]
3. **Finding points of intersection:**
To see if the graphs intersect at \( (1, 2) \):
- Calculate \( f(1) \) and \( g(1) \):
- \( f(1) = 1^3 + 1 = 2 \)
- \( g(1) = 1 + 1 = 2 \)
So, both functions do intersect at \( (1, 2) \).
Now, let's summarize the intercepts:
- Both functions have:
- X-intercept: \((-1, 0)\)
- Y-intercept: \((0, 1)\)
The graphs intersect at \((1, 2)\).
Now we can analyze the given responses:
- "Both functions have an x-intercept of \((-1, 0)\) and a y-intercept of \((0, 1)\)." (True)
- "The graphs intersect at \((1, 2)\)." (True)
- "The x- and y-intercepts are different for each function." (False)
- "Both functions have a y-intercept of \((-1, 0)\) and an x-intercept of \((0, 1)\)" (False)
Since the first two statements are both true based on the analysis, and depending on the context of the question where only one statement should be chosen, you may choose the one that emphasizes the intercepts specifically. Both functions have the same intercepts, and they intersect at \((1, 2)\).
Generally, the statement "Both functions have an x-intercept of \((-1, 0)\) and a y-intercept of \((0, 1)\)." would be the most comprehensive one, but since the statement "The graphs intersect at \((1, 2)\)." is also true and focuses on their behavior, it may also work depending on the question's intent.
If you need to choose one, it could be either:
1. "Both functions have an x-intercept of \((-1, 0)\) and a y-intercept of \((0, 1)\)."
2. "The graphs intersect at \((1, 2)\)."
Either can be deemed correct depending on what the question focuses on.
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