To analyze the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), we first need to determine the x-intercepts (where \( f(x) = 0 \) or \( g(x) = 0 \)) and the y-intercepts (where the functions are evaluated at \( x = 0 \)).
Finding the y-intercepts:
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For \( f(x) = x^3 + 1 \):
- \( f(0) = 0^3 + 1 = 1 \)
- Y-intercept: \( (0, 1) \)
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For \( g(x) = x + 1 \):
- \( g(0) = 0 + 1 = 1 \)
- Y-intercept: \( (0, 1) \)
Finding the x-intercepts:
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For \( f(x) = x^3 + 1 \):
- Set \( f(x) = 0 \): \( x^3 + 1 = 0 \) \( x^3 = -1 \) \( x = -1 \)
- X-intercept: \( (-1, 0) \)
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For \( g(x) = x + 1 \):
- Set \( g(x) = 0 \): \( x + 1 = 0 \) \( x = -1 \)
- X-intercept: \( (-1, 0) \)
Summary of Intercepts:
- Both functions have the same y-intercept: \( (0, 1) \).
- Both functions have the same x-intercept: \( (-1, 0) \).
Comparison of Statements:
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The x- and y-intercepts are different for each function.
- False
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The graphs intersect at \( (1,2) \).
- True; at \( x = 1 \), \( f(1) = 2 \) and \( g(1) = 2 \), they intersect at \( (1, 2) \).
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Both functions have a y-intercept of \( (-1,0) \) and an x-intercept of \( (0,1) \).
- False (intercepts are incorrectly stated).
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Both functions have an x-intercept of \( (-1,0) \) and a y-intercept of \( (0,1) \).
- True (this is precisely correct).
From the options, the true statements are:
- The graphs intersect at \( (1, 2) \).
- Both functions have an x-intercept of \( (-1,0) \) and a y-intercept of \( (0,1) \).
Since the prompt may only want to select one option, the more direct answers are the statements regarding the intercepts:
Final answer: Both functions have an x-intercept of \( (-1,0) \) and a y-intercept of \( (0,1) \).
1 answer