To find the equation of the transformed function [tex]\( g(x) \)[/tex] when the original function [tex]\( f(x) \)[/tex] undergoes a horizontal compression, we need to follow certain steps. The function we start with is:
[tex]\[ f(x) = 7^x + 1 \][/tex]
### Step-by-Step Solution:
1. Understand Horizontal Compression: A horizontal compression by a factor of [tex]\( \frac{1}{3} \)[/tex] means that the [tex]\( x \)[/tex]-values are multiplied by 3. This is because, in the general transformation [tex]\( f(bx) \)[/tex], if [tex]\( b > 1 \)[/tex], the function is compressed horizontally by the factor [tex]\( \frac{1}{b} \)[/tex].
2. Apply the Transformation: To compress the function [tex]\( f(x) \)[/tex] horizontally by a factor of [tex]\( \frac{1}{3} \)[/tex], we replace [tex]\( x \)[/tex] with [tex]\( 3x \)[/tex]. This process modifies the x-coordinate to achieve the desired compression.
3. Rewrite the Function:
[tex]\[ g(x) = f(3x) \][/tex]
So, substituting [tex]\( 3x \)[/tex] in place of [tex]\( x \)[/tex] in the original function [tex]\( f(x) \)[/tex], we get:
[tex]\[ g(x) = 7^{3x} + 1 \][/tex]
### Conclusion:
Hence, the equation of the transformed function [tex]\( g \)[/tex] is:
[tex]\[ g(x) = (7)^{3x} + 1 \][/tex]
This concludes the step-by-step transformation of the original function under the given horizontal compression.
