Question

Graph and compare f(x)=(x+2)2
and g(x)=(12x+1)3
. What point do both functions share?(1 point)
Responses

The functions do not share any points.
The functions do not share any points.

Both functions have the same y-intercept at (0,1)
.
Both functions have the same y -intercept at left parenthesis 0 comma 1 right parenthesis .

Both functions have the same y-intercept at (0,−2)
.
Both functions have the same y -intercept at left parenthesis 0 comma negative 2 right parenthesis .

Both functions have the same x-intercept at (−2,0)
.

Answer 1

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 \((-2,0)\).**