To compare the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \) in terms of their intercepts, let's calculate the x-intercepts and y-intercepts for both functions.
Finding the intercepts
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X-Intercepts:
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For \( f(x) = 0 \): \[ x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1. \] So, the x-intercept of \( f(x) \) is \( (-1, 0) \).
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For \( g(x) = 0 \): \[ x + 1 = 0 \implies x = -1. \] So, the x-intercept of \( g(x) \) is also \( (-1, 0) \).
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Y-Intercepts:
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For \( f(x) \), substitute \( x = 0 \): \[ f(0) = 0^3 + 1 = 1. \] So, the y-intercept of \( f(x) \) is \( (0, 1) \).
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For \( g(x) \), substitute \( x = 0 \): \[ g(0) = 0 + 1 = 1. \] So, the y-intercept of \( g(x) \) is also \( (0, 1) \).
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Conclusion about the intercepts
- Both functions \( f(x) \) and \( g(x) \) share the same x-intercept of \( (-1, 0) \) and the same y-intercept of \( (0, 1) \).
Checking the statements:
- a. both functions have an x-intercept of (-1, 0) and a y-intercept of (0, 1) - True.
- b. both functions have a y-intercept of (-1, 0) and a x-intercept of (0, 1) - False (the x- and y-intercepts are incorrectly stated).
- c. the x- and y-intercepts are different for each function. - False (they are the same).
- d. the graphs intersect at (1, 2) - False (we have not determined this intersection yet).
Finding intersection point (optional)
To find if they intersect at (1, 2), we check: \[ f(1) = 1^3 + 1 = 2 \quad \text{and} \quad g(1) = 1 + 1 = 2. \] So the point (1, 2) is indeed on both graphs. However, this does not change the fact about the intercepts.
Final Answer:
The correct statement is a. both functions have an x-intercept of (-1, 0) and a y-intercept of (0, 1).
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