Find an explicit formula for \( f^{-1} \), Use this formula to graph \( f^{-1}, f \), and the line \( y=x \) on the same screen. To check your work, see whether the graphs of \( f \) and \( f-1 \) are reflections about the line. \[ f(x)=\sqrt{5 x+2} \] \( f^{-1}(x)=\square \)

To find the inverse of the function \( f(x) = \sqrt{5x + 2} \), we start by replacing \( f(x) \) with \( y \): \[ y = \sqrt{5x + 2} \] Next, we swap \( x \) and \( y \): \[ x = \sqrt{5y + 2} \] Now, we square both sides to eliminate the square root: \[ x^2 = 5y + 2 \] Solving for \( y \) gives us: \[ 5y = x^2 - 2 \quad \Rightarrow \quad y = \frac{x^2 - 2}{5} \] Thus, the explicit formula for the inverse function is: \[ f^{-1}(x) = \frac{x^2 - 2}{5} \] Now you can graph \( f(x) \), \( f^{-1}(x) \), and the line \( y = x \) on the same coordinate plane. The graphs of \( f \) and \( f^{-1} \) should indeed reflect over the line \( y = x \), confirming that you've properly found the inverse. Remember, when graphing, \( f(x) \) will lie above the line \( y = x \) for \( x \geq -\frac{2}{5} \), while \( f^{-1}(x) \) will be below it for \( y \) values generated from \( f(x) \).