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For each function below, find \( f^{-1}(x) \) \( \begin{array}{ll}\text { 13. } f(x)=x+3 & \text { 14. } f(x)=x+5 \\ \text { 15. } f(x)=2-x & \text { 16. } f(x)=3-x \\ \text { 17. } f(x)=11 x+7 & \text { 18. } f(x)=9+10 x\end{array} \)

Ask by Mullins Bob. in the United States
Mar 05,2025

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Answer

The inverse functions are: 13. \( f^{-1}(x) = x - 3 \) 14. \( f^{-1}(x) = x - 5 \) 15. \( f^{-1}(x) = 2 - x \) 16. \( f^{-1}(x) = 3 - x \) 17. \( f^{-1}(x) = \frac{x - 7}{11} \) 18. \( f^{-1}(x) = \frac{x - 9}{10} \)

Solution

Below is the step‐by‐step process to find the inverse function \( f^{-1}(x) \) for each given function. --- ### 13. \( f(x)=x+3 \) 1. Write the function as: \[ y = x+3. \] 2. Swap \( x \) and \( y \): \[ x = y+3. \] 3. Solve for \( y \): \[ y = x-3. \] Thus, the inverse function is: \[ f^{-1}(x)=x-3. \] --- ### 14. \( f(x)=x+5 \) 1. Write the function as: \[ y = x+5. \] 2. Swap \( x \) and \( y \): \[ x = y+5. \] 3. Solve for \( y \): \[ y = x-5. \] Thus, the inverse function is: \[ f^{-1}(x)=x-5. \] --- ### 15. \( f(x)=2-x \) 1. Write the function as: \[ y = 2-x. \] 2. Swap \( x \) and \( y \): \[ x = 2-y. \] 3. Solve for \( y \): \[ y = 2-x. \] Thus, the inverse function is: \[ f^{-1}(x)=2-x. \] --- ### 16. \( f(x)=3-x \) 1. Write the function as: \[ y = 3-x. \] 2. Swap \( x \) and \( y \): \[ x = 3-y. \] 3. Solve for \( y \): \[ y = 3-x. \] Thus, the inverse function is: \[ f^{-1}(x)=3-x. \] --- ### 17. \( f(x)=11x+7 \) 1. Write the function as: \[ y = 11x+7. \] 2. Swap \( x \) and \( y \): \[ x = 11y+7. \] 3. Isolate the term with \( y \): \[ 11y = x-7. \] 4. Solve for \( y \): \[ y = \frac{x-7}{11}. \] Thus, the inverse function is: \[ f^{-1}(x)=\frac{x-7}{11}. \] --- ### 18. \( f(x)=9+10x \) 1. Write the function as: \[ y = 9+10x. \] 2. Swap \( x \) and \( y \): \[ x = 9+10y. \] 3. Isolate the term with \( y \): \[ 10y = x-9. \] 4. Solve for \( y \): \[ y = \frac{x-9}{10}. \] Thus, the inverse function is: \[ f^{-1}(x)=\frac{x-9}{10}. \] --- Each function’s inverse was determined by swapping \( x \) and \( y \) and then solving for \( y \).

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Bonus Knowledge

To find the inverse functions for each of the functions provided, we will follow these steps: replace \( f(x) \) with \( y \), swap \( x \) and \( y \), and then solve for \( y \). 1. **For \( f(x) = x + 3 \)**: - Set \( y = x + 3 \) - Swap: \( x = y + 3 \) - Solve for \( y \): \( y = x - 3 \) - Thus, \( f^{-1}(x) = x - 3 \) 2. **For \( f(x) = x + 5 \)**: - Set \( y = x + 5 \) - Swap: \( x = y + 5 \) - Solve for \( y \): \( y = x - 5 \) - Thus, \( f^{-1}(x) = x - 5 \) 3. **For \( f(x) = 2 - x \)**: - Set \( y = 2 - x \) - Swap: \( x = 2 - y \) - Solve for \( y \): \( y = 2 - x \) - Thus, \( f^{-1}(x) = 2 - x \) 4. **For \( f(x) = 3 - x \)**: - Set \( y = 3 - x \) - Swap: \( x = 3 - y \) - Solve for \( y \): \( y = 3 - x \) - Thus, \( f^{-1}(x) = 3 - x \) 5. **For \( f(x) = 11x + 7 \)**: - Set \( y = 11x + 7 \) - Swap: \( x = 11y + 7 \) - Solve for \( y \): \( y = \frac{x - 7}{11} \) - Thus, \( f^{-1}(x) = \frac{x - 7}{11} \) 6. **For \( f(x) = 9 + 10x \)**: - Set \( y = 9 + 10x \) - Swap: \( x = 9 + 10y \) - Solve for \( y \): \( y = \frac{x - 9}{10} \) - Thus, \( f^{-1}(x) = \frac{x - 9}{10} \) So, the final inverses are: - \( f^{-1}(x) = x - 3 \) for \( f(x) = x + 3 \) - \( f^{-1}(x) = x - 5 \) for \( f(x) = x + 5 \) - \( f^{-1}(x) = 2 - x \) for \( f(x) = 2 - x \) - \( f^{-1}(x) = 3 - x \) for \( f(x) = 3 - x \) - \( f^{-1}(x) = \frac{x - 7}{11} \) for \( f(x) = 11x + 7 \) - \( f^{-1}(x) = \frac{x - 9}{10} \) for \( f(x) = 9 + 10x \)

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