The image provided contains the following mathematical problem in Indonesian:

"Diketahui fungsi $f(x) = 2^{x-1} + 5$. Jika fungsi $f(x)$ ditranslasikan oleh $T \left( \begin{array}{c} 4 \\ -1 \end{array} \right)$, tentukanlah hasil translasi nya."

Translation of the question:

"Given the function $f(x) = 2^{x-1} + 5$. If the function $f(x)$ is translated by the transformation $T \left( \begin{array}{c} 4 \\ -1 \end{array} \right)$, determine the result of the translation."

Solution:

The translation described here is a horizontal and vertical shift. The transformation vector $T \left( \begin{array}{c} 4 \\ -1 \end{array} \right)$ indicates:

• A horizontal shift of 4 units to the right.
• A vertical shift of -1 unit downwards.

Step 1: Horizontal Shift

The horizontal translation of 4 units to the right affects the $x$-values. To reflect this, replace $x$ with $x - 4$ in the function.

Thus, the new function after horizontal translation is: $f(x - 4) = 2^{(x-4) - 1} + 5 = 2^{x - 5} + 5$

Step 2: Vertical Shift

The vertical translation of -1 unit affects the $y$-values. To reflect this, subtract 1 from the entire function.

Thus, the new function after both translations is: $f_{\text{translated}}(x) = 2^{x-5} + 5 - 1 = 2^{x-5} + 4$

Final Answer:

The translated function is $f_{\text{translated}}(x) = 2^{x-5} + 4$.

Would you like further details or explanations?

Here are 5 related questions to consider:

1. What is the general formula for translating a function?
2. How do horizontal and vertical shifts affect the graph of exponential functions?
3. How would the graph of $f(x) = 2^{x-1} + 5$ look before and after translation?
4. What is the effect of shifting a function by a transformation vector $T \left( \begin{array}{c} a \\ b \end{array} \right)$?
5. How can you apply transformations to other types of functions, such as quadratic or trigonometric functions?

Tip: When applying a horizontal shift, remember that replacing $x$ with $x - a$ shifts the graph right by $a$ units, and replacing $x$ with $x + a$ shifts it left by $a$ units.