To analyze the transformation from the equation \( f(x) = 4^x - 5 \) to \( 4^{2x} - 5 \), we can observe how the exponent is modified.
When \( f(x) = 4^x - 5 \) is transformed to \( f(x) = 4^{2x} - 5 \), the exponent \( x \) is replaced by \( 2x \). This indicates that the input to the function is scaled by a factor of 2.
In general, when the exponent of a function \( a^x \) is multiplied by a factor (\( k = 2 \) in this case), it results in a horizontal compression of the graph by a factor of \( 1/k \). Therefore, because the exponent of \( x \) is doubled (i.e., \( 2x \)), this graph undergoes a horizontal compression.
Based on this analysis, we can conclude:
- The value of \( k \) is 2.
- The graph shows a compression.
So, the correct response is:
2; compression.