Compare the functions f(x) = x ^ 3 + 1 and g(x) = x + 1 Which of the following statements is true about the intercepts of the functions
The intercepts of a function are the points at which the function intersects the x-axis or the y-axis.
For the function f(x) = x^3 + 1:
- To find the x-intercept, we set y (or f(x)) equal to 0 and solve for x:
0 = x^3 + 1
This equation does not have any real solutions, so there is no x-intercept.
- The y-intercept occurs when x is equal to 0:
f(0) = 0^3 + 1 = 1
Therefore, the y-intercept is at (0, 1).
For the function g(x) = x + 1:
- To find the x-intercept, we set y (or g(x)) equal to 0 and solve for x:
0 = x + 1
x = -1
Therefore, the x-intercept is at (-1, 0).
- The y-intercept occurs when x is equal to 0:
g(0) = 0 + 1 = 1
Therefore, the y-intercept is at (0, 1).
From the above analysis, we can conclude that both functions f(x) = x^3 + 1 and g(x) = x + 1 have the same y-intercept at (0, 1), but they have different x-intercepts. In particular, f(x) = x^3 + 1 has no x-intercept, while g(x) = x + 1 has an x-intercept at (-1, 0).