To analyze the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \), we need to identify their intercepts.
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Finding the y-intercept:
- The y-intercept occurs where \( x = 0 \).
- For \( f(0) = 0^3 + 1 = 1 \) → y-intercept is \( (0, 1) \).
- For \( g(0) = 0 + 1 = 1 \) → y-intercept is \( (0, 1) \).
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Finding the x-intercept:
- The x-intercept occurs where \( f(x) = 0 \) and \( g(x) = 0 \).
- For \( f(x) = 0 \): \( x^3 + 1 = 0 \) → \( x^3 = -1 \) → \( x = -1 \) → x-intercept is \( (-1, 0) \).
- For \( g(x) = 0 \): \( x + 1 = 0 \) → \( x = -1 \) → x-intercept is \( (-1, 0) \).
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Checking the intersection points:
- To find where the graphs intersect, set \( f(x) = g(x) \): \[ x^3 + 1 = x + 1 \] \[ x^3 = x \] \[ x(x^2 - 1) = 0 \] This gives us \( x = 0 \) or \( x^2 = 1 \) (so \( x = 1 \) or \( x = -1 \)).
- Evaluating these:
- For \( x = -1 \): \( f(-1) = 0 \) and \( g(-1) = 0 \) → they intersect at \( (-1, 0) \).
- For \( x = 0 \): \( f(0) = 1 \) and \( g(0) = 1 \) → they intersect at \( (0, 1) \).
- For \( x = 1 \): \( f(1) = 2 \) and \( g(1) = 2 \) → they intersect at \( (1, 2) \).
Now evaluating the statements:
- Statement 1: "Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1)." – False.
- Statement 2: "The graphs intersect at (1, 2)." – True.
- Statement 3: "The x- and y-intercepts are different for each function." – False (they share x- and y-intercepts).
- Statement 4: "Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1)." – True, but only the x-intercept statement is correct since they have the same intercepts.
Thus, the true statements are that "The graphs intersect at (1,2)" and "Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1)".
The best response based on the question is:
The graphs intersect at (1, 2).