https://file.aiquickdraw.com/mathbot/file/1726165139204sjk0iota.jpg

The function provided is:

\[
f(x) = \left(\frac{1}{6}\right)^x
\]

We are told that the function is translated 7 units to the left to create \( g(x) \). A translation to the left affects the input variable \(x\) by adding the translation value to it.

So, when translating 7 units left, the new function becomes:

\[
g(x) = f(x + 7)
\]

Substituting \( f(x) = \left(\frac{1}{6}\right)^x \) into the equation for \( g(x) \), we get:

\[
g(x) = \left(\frac{1}{6}\right)^{x + 7}
\]

Thus, the equation for \( g(x) \) is:

\[
g(x) = \left(\frac{1}{6}\right)^{x + 7}
\]

Let me know if you'd like more details or further explanations!

Here are some related questions you might find helpful:
1. What is the general rule for translating functions horizontally?
2. How would translating a function to the right affect the equation?
3. How do vertical translations differ from horizontal translations?
4. What is the base of an exponential function and how does it affect its graph?
5. How does the graph of \( g(x) = \left(\frac{1}{6}\right)^{x+7} \) differ from \( f(x) = \left(\frac{1}{6}\right)^x \)?

**Tip:** When translating a function horizontally, remember to adjust the input variable \(x\) inside the function, adding for leftward shifts and subtracting for rightward shifts.

The function f(x) = (1/6)^x is translated 7 units left to create g(x). What is an equation of g(x)?

Learn how to translate the exponential function f(x) = (1/6)^x 7 units to the left, resulting in the function g(x) = (1/6)^(x + 7). This problem explores horizontal translations in functions and exponential function properties, suitable for Grades 9-12 students.

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