Let's analyze the intercepts of the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \).
For the function \( f(x) = x^3 + 1 \):
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Y-intercept: This occurs when \( x = 0 \): \[ f(0) = 0^3 + 1 = 1 \] So the y-intercept is \( (0, 1) \).
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X-intercept: This occurs when \( f(x) = 0 \): \[ x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1 \] So the x-intercept is \( (-1, 0) \).
For the function \( g(x) = x + 1 \):
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Y-intercept: This occurs when \( x = 0 \): \[ g(0) = 0 + 1 = 1 \] So the y-intercept is also \( (0, 1) \).
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X-intercept: This occurs when \( g(x) = 0 \): \[ x + 1 = 0 \implies x = -1 \] So the x-intercept is \( (-1, 0) \).
Summary of intercepts:
- Both functions have the same y-intercept \( (0, 1) \).
- Both functions have the same x-intercept \( (-1, 0) \).
Given this analysis, let's evaluate the provided statements:
- 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)
- Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1). (True)
- The graphs intersect at (1,2). (False)
Therefore, the true statement is:
Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1).