To find the equation of the transformed function [tex]\(g\)[/tex] given the original function [tex]\(f(x) = 2^x - 1\)[/tex] and a horizontal shift of 7 units to the left, we follow these steps:
1. Identify the original function: [tex]\( f(x) = 2^x - 1 \)[/tex].
2. Determine the horizontal shift of 7 units to the left. A horizontal shift to the left by 7 units can be accounted for by replacing [tex]\(x\)[/tex] in the original function with [tex]\(x + 7\)[/tex].
3. Recognize that the given form for the transformed function is [tex]\( g(x) = 2^{x + h} + k \)[/tex].
4. Since the function is shifted 7 units to the left, [tex]\(h\)[/tex] will be [tex]\(-7\)[/tex]: [tex]\( g(x) = 2^{x + (-7)} + k \)[/tex].
5. The original function [tex]\(f(x)\)[/tex] has a vertical translation of [tex]\(-1\)[/tex], which means [tex]\(k = -1\)[/tex].
6. Substitute the values of [tex]\(h\)[/tex] and [tex]\(k\)[/tex] into the given transformed function form:
[tex]\[ g(x) = 2^{x + (-7)} + (-1) \][/tex]
Simplifying, we get:
[tex]\[ g(x) = 2^{x - 7} - 1 \][/tex]
Hence, the equation of the transformed function [tex]\(g\)[/tex] is:
[tex]\[ g(x) = 2^{x - 7} - 1 \][/tex]
