5.2.2 If $$ k(x)=-f(x) $$, write down the equation of $$ k $$. 5.2.3 If $$ f(x)=3^{x-1}+2 $$ is translated to $$ t $$ such that $$ t(x)= $$ (a) Determine $$ t^{-1} $$ in the form of $$ y=\ldots $$ (b) Describe the transformation from $$ f $$ to $$ t $$.

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### **5.2.2 If $$ k(x) = -f(x) $$, write down the equation of $$ k $$.**

**Answer:**

Given that $$ k(x) $$ is defined as the negative of $$ f(x) $$, the equation for $$ k(x) $$ is simply:

$$
k(x) = -f(x)
$$

*This means that for every value of $$ x $$, $$ k(x) $$ is the opposite of $$ f(x) $$. If $$ f(x) $$ outputs a positive value, $$ k(x) $$ will output a negative value of the same magnitude, and vice versa.*

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### **5.2.3 If $$ f(x) = 3^{x-1} + 2 $$ is translated to $$ t $$ such that $$ t(x) = \ldots $$**

Let's assume that the translation involves transforming $$ f(x) $$ to $$ t(x) $$ by shifting it horizontally and/or vertically. Without additional specifics about the translation, we'll consider the standard form provided.

#### **(a) Determine $$ t^{-1} $$ in the form of $$ y = \ldots $$**

**Answer:**

To find the inverse function $$ t^{-1}(x) $$, we need to solve for $$ y $$ in terms of $$ x $$ such that:

$$
y = t(x) = 3^{x-1} + 2
$$

**Steps to Find the Inverse:**

1. **Start with the original equation:**
   $$
   y = 3^{x-1} + 2
   $$

2. **Subtract 2 from both sides:**
   $$
   y - 2 = 3^{x-1}
   $$

3. **Apply the logarithm with base 3 to both sides to solve for $$ x $$:**
   $$
   \log_3(y - 2) = x - 1
   $$

4. **Add 1 to both sides to isolate $$ x $$:**
   $$
   x = \log_3(y - 2) + 1
   $$

5. **Swap $$ x $$ and $$ y $$ to express the inverse function:**
   $$
   t^{-1}(x) = \log_3(x - 2) + 1
   $$

**So, the inverse function is:**

$$
t^{-1}(x) = \log_3(x - 2) + 1
$$

#### **(b) Describe the transformation from $$ f $$ to $$ t $$.**

**Answer:**

Given the original function $$ f(x) = 3^{x-1} + 2 $$, the transformation to $$ t(x) $$ involves the following steps:

1. **Horizontal Shift:**
   - The term $$ x - 1 $$ inside the exponent indicates a **horizontal shift to the right by 1 unit**. This means that the graph of the exponential function $$ 3^x $$ is shifted rightward along the x-axis.

2. **Vertical Shift:**
   - The $$ +2 $$ outside the exponent signifies a **vertical shift upward by 2 units**. This moves the entire graph of the function upward along the y-axis.

**Summary of the Transformation:**
- **Right Shift:** 1 unit
- **Upward Shift:** 2 units

*These combined shifts result in the graph of $$ t(x) $$ being positioned 1 unit to the right and 2 units above the original graph of $$ 3^x $$.*

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### **Visual Representation**

To better understand the transformation, here's a simplified visual representation:

- **Original Function $$ 3^x $$:** Starts at $$ (0, 1) $$.
- **After Shifting Right by 1:** Becomes $$ 3^{x-1} $$, starting at $$ (1, 1) $$.
- **After Shifting Up by 2:** Becomes $$ 3^{x-1} + 2 $$, starting at $$ (1, 3) $$.

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Feel free to ask if you need further clarification on any of these steps!
**5.2.2** $$ k(x) = -f(x) $$ **5.2.3 (a)** $$ t^{-1}(x) = \log_3(x - 2) + 1 $$ **5.2.3 (b)** The function $$ t(x) $$ is obtained by shifting $$ f(x) = 3^{x-1} + 2 $$ **1 unit to the right** and **2 units up**.

5.2.2 If \( k(x)=-f(x) \), write down the equation of \( k \). 5.2.3 If \( f(x)=3^{x-1}+2 \) is translated to \( t \) such that \( t(x)= \) (a) Determine \( t^{-1} \) in the form of \( y=\ldots \) (b) Describe the transformation from \( f \) to \( t \).

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