To find the shared points of the functions \( f(x) = (x+2)^2 \) and \( g(x) = (12x+1)^3 \), we will examine both functions, specifically looking for their y-intercepts and x-intercepts.
1. Finding the y-intercepts:
-
For \( f(x) \): \[ f(0) = (0 + 2)^2 = 2^2 = 4 \] So, the y-intercept of \( f(x) \) is at \( (0, 4) \).
-
For \( g(x) \): \[ g(0) = (12 \cdot 0 + 1)^3 = 1^3 = 1 \] So, the y-intercept of \( g(x) \) is at \( (0, 1) \).
2. Finding the x-intercepts:
-
For \( f(x) \): Set \( f(x) = 0 \): \[ (x+2)^2 = 0 \implies x + 2 = 0 \implies x = -2. \] So, the x-intercept of \( f(x) \) is at \( (-2, 0) \).
-
For \( g(x) \): Set \( g(x) = 0 \): \[ (12x + 1)^3 = 0 \implies 12x + 1 = 0 \implies 12x = -1 \implies x = -\frac{1}{12}. \] So, the x-intercept of \( g(x) \) is at \( \left(-\frac{1}{12}, 0\right) \).
Summary of Intercepts:
- \( f(x) \) has y-intercept at \( (0, 4) \) and x-intercept at \( (-2, 0) \).
- \( g(x) \) has y-intercept at \( (0, 1) \) and x-intercept at \( \left(-\frac{1}{12}, 0\right) \).
Conclusion:
The only shared point between both functions is the x-intercept at \( (-2, 0) \).
Answer: Both functions have the same x-intercept at \((-2,0)\).