To determine whether the function \( f(x) = \sqrt{x} + 3 + 8 \) and \( f(4x) = \sqrt{4x} + 3 + 8 \) represents a compression or stretch, we can look at how the argument of the function changes.
The function \( f(4x) \) can be rewritten as follows: \[ f(4x) = \sqrt{4x} + 3 + 8 = 2\sqrt{x} + 3 + 8 \] This shows that for every \( x \) in the transformed function \( f(4x) \), the output is dependent on \( \sqrt{4x} \) instead of just \( \sqrt{x} \).
When the input to the function is scaled by a factor of 4 (that is, replaced by \( 4x \)), it affects how the graph stretches or compresses. Specifically, scaling \( x \) by a factor greater than 1 causes a horizontal compression of the graph. Therefore, since we have \( 4x \) as the argument in the second function:
- Compression occurs when the argument is multiplied by a factor greater than \( 1\).
Hence, the modification from \( x \) to \( 4x \) indicates a compression in the graph of the function.
Therefore, the answer is:
1 (for compression).