https://file.aiquickdraw.com/mathbot/file/1727209795279h1d7z4sd.jpg

The problem in the image asks about translating the function \( f(x) = 2^{3x - 1} \) by the translation vector \( T = \begin{pmatrix} 4 \\ -1 \end{pmatrix} \).

To solve this, we need to understand how translation works on a function. A translation by \( T = \begin{pmatrix} a \\ b \end{pmatrix} \) affects the function as follows:
- The horizontal translation \( a \) shifts the input \( x \) by \( -a \), so the new input becomes \( x - a \).
- The vertical translation \( b \) adds \( b \) to the function's output.

Given that \( T = \begin{pmatrix} 4 \\ -1 \end{pmatrix} \), the function undergoes the following changes:
- Horizontal shift by \( -4 \), which modifies \( x \) as \( x - 4 \).
- Vertical shift by \( -1 \), which subtracts 1 from the function's output.

### Step 1: Apply Horizontal Shift
We replace \( x \) with \( x - 4 \) in the function:
\[
f(x - 4) = 2^{3(x - 4) - 1} = 2^{3x - 12 - 1} = 2^{3x - 13}.
\]

### Step 2: Apply Vertical Shift
Now we subtract 1 from the result to account for the vertical translation:
\[
f_T(x) = 2^{3x - 13} - 1.
\]

Thus, the translated function is:
\[
f_T(x) = 2^{3x - 13} - 1.
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you determine the direction and magnitude of a translation vector?
2. How would the function change if the translation vector was \( T = \begin{pmatrix} -2 \\ 3 \end{pmatrix} \)?
3. Can you translate other types of functions, like quadratic or trigonometric functions, in a similar way?
4. How does horizontal and vertical translation affect the graph of the function?
5. How would the solution change if the function were \( f(x) = 3x^2 \) instead of an exponential?

**Tip:** Always remember that a horizontal translation modifies the input \( x \), while a vertical translation affects the output of the function directly.

Diketahui fungsi f(x) = 2^(3x-1). Jika fungsi ditranslasikan oleh T = (4, -1), hasilnya adalah ...

Learn how to translate the exponential function f(x) = 2^(3x - 1) by the translation vector T = (4, -1). Understand the horizontal and vertical shifts applied to the function and the resulting transformed function.

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