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The function [tex]$f(x)=2^x-1$[/tex] is transformed to function [tex]$g$[/tex] through a horizontal shift of 7 units left. What is the equation of function [tex]$g$[/tex]? Replace the values of [tex]$h$[/tex] and [tex]$k$[/tex] in the equation.
[tex]$g(x)=2^{x+h}+k$[/tex]
Answer (2)
Apply a horizontal shift of 7 units left to the function f ( x ) = 2 x − 1 by replacing x with x + 7 .
Obtain the transformed function g ( x ) = 2 x + 7 − 1 .
Compare g ( x ) with the given form g ( x ) = 2 x + h + k to identify h = 7 and k = − 1 .
The equation of the transformed function is g ( x ) = 2 x + 7 − 1 .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 x − 1 and asked to find the equation of the function g ( x ) after a horizontal shift of 7 units to the left. The general form of the transformed function is given as g ( x ) = 2 x + h + k , and our goal is to find the values of h and k .
Applying the Horizontal Shift To shift the function f ( x ) horizontally by 7 units to the left, we replace x with x + 7 in the expression for f ( x ) . This gives us:
g ( x ) = f ( x + 7 ) = 2 x + 7 − 1
Identifying h and k Now we compare the transformed function g ( x ) = 2 x + 7 − 1 with the given form g ( x ) = 2 x + h + k . By matching the terms, we can identify the values of h and k . We have:
2 x + h = 2 x + 7 , which implies h = 7 .
And k = − 1 .
Final Equation Therefore, the equation of the transformed function g ( x ) is:
g ( x ) = 2 x + 7 − 1
Examples
Horizontal shifts of functions are used in various fields, such as signal processing and image analysis. For example, in signal processing, a time delay can be represented as a horizontal shift of a signal. In image analysis, shifting an image horizontally can be used for alignment or creating special effects. Understanding how to perform and analyze these transformations is crucial in these applications.
To find the function after a horizontal shift, replace x with x + 7 in f ( x ) = 2 x − 1 , resulting in g ( x ) = 2 x + 7 − 1 . Thus, the values are h = 7 and k = − 1 . The final equation is g ( x ) = 2 x + 7 − 1 .
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