The function $f(x)=7^x+1$ is transformed to function $g$ through a horizontal compression by a factor of $\frac{1}{3}$. What is the equation of function $g$?
$g(x)=(7)^{k x}+1$
Answer (2)
The original function is f ( x ) = 7 x + 1 .
A horizontal compression by a factor of 3 1 means replacing x with 3 x .
The transformed function is g ( x ) = f ( 3 x ) = 7 3 x + 1 .
Therefore, the equation of the transformed function is g ( x ) = 7 3 x + 1 , and k = 3 , so the answer is 3 .
Explanation
Understanding the Transformation The problem states that the function f ( x ) = 7 x + 1 is transformed into a function g ( x ) by a horizontal compression by a factor of 3 1 . We need to find the equation of the transformed function g ( x ) . A horizontal compression by a factor of 3 1 means that the input x is replaced by 3 x .
Applying the Horizontal Compression To find the transformed function g ( x ) , we replace x with 3 x in the original function f ( x ) . So, we have g ( x ) = f ( 3 x ) = 7 3 x + 1
Finding the Value of k The transformed function is g ( x ) = 7 3 x + 1 . Comparing this with the given form g ( x ) = ( 7 ) k x + 1 , we can identify the value of k .
g ( x ) = ( 7 ) k x + 1 = 7 3 x + 1 Therefore, k = 3 .
Final Answer The equation of the transformed function g ( x ) is g ( x ) = 7 3 x + 1 . The value of k is 3.
Examples
Horizontal compression and stretching of functions are used in image processing to resize images. For example, if you want to compress an image horizontally to fit a smaller screen, you are essentially applying a horizontal compression to the function that represents the image. Similarly, stretching an audio signal horizontally changes its playback speed, which is a real-world application of horizontal transformations.
The equation of the transformed function g ( x ) after a horizontal compression by a factor of 3 1 from f ( x ) = 7 x + 1 is g ( x ) = 7 3 x + 1 , where k = 3 .
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