7. Given the function \( f(x)=(x+4)^{2}-1 \), a) determine the inverse of the function. Identify a restricted domain for which the function has an inverse that is also a function b) graph the function and its inverse on the same set of axes

To find the inverse of the function \( f(x) = (x + 4)^2 - 1 \), we will follow these steps: ### Part a: Determine the Inverse 1. **Identify the function**: \[ y = (x + 4)^2 - 1 \] 2. **Solve for \( x \)** in terms of \( y \): \[ y + 1 = (x + 4)^2 \] \[ \sqrt{y + 1} = |x + 4| \] Since we want to find the inverse, we need to consider the restricted domain. The function \( f(x) \) is a parabola that opens upwards, and it is symmetric about its vertex. The vertex of the parabola is at \( x = -4 \). To ensure that the function is one-to-one, we can restrict the domain to \( x \geq -4 \). Thus, we can drop the absolute value: \[ \sqrt{y + 1} = x + 4 \] \[ x = \sqrt{y + 1} - 4 \] 3. **Express the inverse function**: \[ f^{-1}(y) = \sqrt{y + 1} - 4 \] 4. **Replace \( y \) with \( x \)** to write the inverse function: \[ f^{-1}(x) = \sqrt{x + 1} - 4 \] ### Restricted Domain The original function \( f(x) \) is restricted to \( x \geq -4 \). The range of \( f(x) \) is \( y \geq -1 \) (since the minimum value occurs at the vertex). Therefore, the inverse function \( f^{-1}(x) \) is defined for \( x \geq -1 \). ### Part b: Graph the Function and Its Inverse 1. **Graph of \( f(x) \)**: - The vertex is at \( (-4, -1) \). - The function opens upwards. - The y-intercept can be found by substituting \( x = 0 \): \[ f(0) = (0 + 4)^2 - 1 = 16 - 1 = 15 \] - The function passes through the point \( (0, 15) \). 2. **Graph of \( f^{-1}(x) \)**: - The inverse function \( f^{-1}(x) = \sqrt{x + 1} - 4 \) has a domain of \( x \geq -1 \). - The inverse function starts at the point \( (-1, 0) \) (since \( f^{-1}(-1) = 0 \)). - The function increases as \( x \) increases. 3. **Plotting**: - The graph of \( f(x) \) is a parabola opening upwards with vertex at \( (-4, -1) \). - The graph of \( f^{-1}(x) \) is a square root function starting at \( (-1, 0) \) and increasing. 4. **Reflection**: - The graph of the inverse function is a reflection of the graph of the original function across the line \( y = x \). ### Summary - The inverse function is \( f^{-1}(x) = \sqrt{x + 1} - 4 \) with a domain of \( x \geq -1 \). - The original function \( f(x) = (x + 4)^2 - 1 \) is restricted to \( x \geq -4 \) to ensure it is one-to-one. You can use graphing software or a graphing calculator to visualize both functions and their reflection across the line \( y = x \).