Question

\begin{aligned}
f(x)&=x^2
\\\\
g(x)&=(x+3)^2+5
\end{aligned}

Answer 1

You've provided two functions, \( f(x) \) and \( g(x) \). Let's look at them more closely:

1. **The function \( f(x) \):**
\[
f(x) = x^2
\]
This is a basic quadratic function, which represents a parabola opening upwards with its vertex at the origin (0,0).

2. **The function \( g(x) \):**
\[
g(x) = (x + 3)^2 + 5
\]
This function is also a quadratic function. The transformation in this function indicates that it is shifted horizontally to the left by 3 units and vertically upwards by 5 units. The vertex of this parabola is at the point (-3, 5).

### Properties of the Functions:
- The vertex of \( f(x) \) is at (0,0).
- The vertex of \( g(x) \) is at (-3,5).
- Both functions are parabolas, but \( g(x) \) is translated from \( f(x) \).

### Evaluating the Functions:
If you'd like to evaluate these functions for a specific value of \( x \) or find their intersection, feel free to ask for those calculations!