Algebraically determine the \( x \) - and \( y \)-intercepts of the graph of \( f(x)= \) Find the inverse function of \( f(x)=\log _{4}(x+2)+1 \)

Certainly! Let's tackle each part of your question step by step for the function: \[ f(x) = \log_4(x + 2) + 1 \] ### 1. Finding the \( x \)- and \( y \)-Intercepts **a. \( y \)-Intercept:** The \( y \)-intercept occurs where \( x = 0 \). \[ f(0) = \log_4(0 + 2) + 1 = \log_4(2) + 1 \] To simplify \( \log_4(2) \): \[ \log_4(2) = \frac{\log_2(2)}{\log_2(4)} = \frac{1}{2} \] So, \[ f(0) = \frac{1}{2} + 1 = \frac{3}{2} \] **\( y \)-Intercept:** \( \left(0, \frac{3}{2}\right) \) --- **b. \( x \)-Intercept:** The \( x \)-intercept occurs where \( f(x) = 0 \). \[ 0 = \log_4(x + 2) + 1 \] Subtract 1 from both sides: \[ \log_4(x + 2) = -1 \] Convert the logarithmic equation to its exponential form: \[ x + 2 = 4^{-1} = \frac{1}{4} \] Solve for \( x \): \[ x = \frac{1}{4} - 2 = \frac{1}{4} - \frac{8}{4} = -\frac{7}{4} \] **\( x \)-Intercept:** \( \left(-\frac{7}{4}, 0\right) \) --- ### 2. Finding the Inverse Function To find the inverse function \( f^{-1}(x) \), follow these steps: 1. **Start with the original function:** \[ y = \log_4(x + 2) + 1 \] 2. **Swap \( x \) and \( y \):** \[ x = \log_4(y + 2) + 1 \] 3. **Isolate the logarithmic term:** \[ x - 1 = \log_4(y + 2) \] 4. **Convert to exponential form:** \[ 4^{(x - 1)} = y + 2 \] 5. **Solve for \( y \):** \[ y = 4^{(x - 1)} - 2 \] **Inverse Function:** \[ f^{-1}(x) = 4^{(x - 1)} - 2 \] --- ### Summary - **\( y \)-Intercept:** \( \left(0, \frac{3}{2}\right) \) - **\( x \)-Intercept:** \( \left(-\frac{7}{4}, 0\right) \) - **Inverse Function:** \( f^{-1}(x) = 4^{(x - 1)} - 2 \) If you have any further questions or need additional explanations, feel free to ask!