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 \( t \).

Sure, let's address each part of your question step by step. --- ### **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.* --- ### **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 \( 3^x \).* --- ### **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 \( (1, 3) \). --- Feel free to ask if you need further clarification on any of these steps!