Which function represents a horizontal compression of an exponential function?
[tex]y=2^x[/tex]
[tex]y=3 \cdot 2^x[/tex]
[tex]y=2^{3 x}[/tex]
Answer (2)
Horizontal compression of a function f ( x ) is given by f ( k x ) , where 1"> k > 1 .
y = 2 x is the original exponential function.
y = 3 c d o t 2 x is a vertical stretch.
y = 2 3 x is a horizontal compression, so the answer is y = 2 3 x .
Explanation
Understanding the Problem We are given three functions: y = 2 x , y = 3 c d o t 2 x , and y = 2 3 x . We need to identify which function represents a horizontal compression of the exponential function y = 2 x .
Horizontal Compression Recall that a horizontal compression of a function y = f ( x ) by a factor of k is given by y = f ( k x ) , where 1"> k > 1 .
Analyzing the Functions The function y = 2 x is the original exponential function. The function y = 3 c d o t 2 x represents a vertical stretch by a factor of 3.
Identifying Horizontal Compression The function y = 2 3 x can be written as y = ( 2 3 ) x = 8 x . This is a horizontal compression of y = 2 x since 2 3 x = 2 ( 3 c d o t x ) , where 1"> k = 3 > 1 .
Conclusion Therefore, the function y = 2 3 x represents a horizontal compression of the exponential function y = 2 x .
Examples
Horizontal compression of functions is used in image processing to reduce the size of an image without losing important details. For example, if you have a wide image that you want to fit into a smaller frame, you can apply horizontal compression to make it fit. This concept is also used in signal processing and data compression to efficiently store and transmit data.
The function that represents a horizontal compression of the exponential function y = 2 x is y = 2 3 x . This is because it follows the transformation rules of horizontal compression by having its input multiplied by a factor greater than 1. Hence, the answer is y = 2 3 x .
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