inverse of the following function? $$ f(x)=(x+7)^{2} $$

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To find the inverse of the function $$ f(x) = (x + 7)^2 $$, we'll follow these steps:

1. **Understand the Function:**
   
   The given function is a quadratic function, which is a parabola. Quadratic functions are not one-to-one over their entire domain because they fail the horizontal line test. However, they can be made one-to-one by restricting their domain.

2. **Restrict the Domain:**
   
   To ensure the function has an inverse, we'll restrict the domain to $$[ -7, \infty )$$. This restriction makes the function increasing and thus one-to-one.

3. **Find the Inverse:**

Start by letting $$ y = f(x) $$:
   $$
   y = (x + 7)^2
   $$
   
   To find the inverse, solve for $$ x $$ in terms of $$ y $$:
   
   $$
   \sqrt{y} = x + 7
   $$
   
   $$
   x = \sqrt{y} - 7
   $$
   
   So, the inverse function is:
   
   $$
   f^{-1}(y) = \sqrt{y} - 7
   $$
   
   Typically, we replace $$ y $$ with $$ x $$ when expressing the inverse function:
   
   $$
   f^{-1}(x) = \sqrt{x} - 7
   $$
   
4. **Domain of the Inverse:**
   
   Since $$ f(x) = (x + 7)^2 $$ maps $$[ -7, \infty )$$ to $$[0, \infty )$$, the inverse function $$ f^{-1}(x) $$ will have a domain of $$[0, \infty )$$.

**Final Answer:**
$$
f^{-1}(x) = \sqrt{x}\, -\, 7
$$
The inverse function is $$ f^{-1}(x) = \sqrt{x} - 7 $$.

inverse of the following function? \[ f(x)=(x+7)^{2} \]

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