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:
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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 \( (0, 1) \).
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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:
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"Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1)."
- False.
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"Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1)."
- False.
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"The x- and y-intercepts are different for each function."
- False.
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"Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1)."
- True.
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"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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