Which function is the inverse of the function f? \[ f(x)=4 x-5 \] A. \( j(x)=\frac{1}{4} x-\frac{1}{5} \) B. \( g(x)=\frac{1}{4} x+\frac{5}{4} \) C. \( k(x)=-4 x+5 \) D. \( h(x)=5 x-4 \)

To find the inverse of the function \( f(x) = 4x - 5 \), we need to switch \( x \) and \( y \) and solve for \( y \). Setting \( y = 4x - 5 \), we swap them to get \( x = 4y - 5 \). Adding 5 to both sides gives \( x + 5 = 4y \), and dividing by 4 results in \( y = \frac{1}{4}(x + 5) \), or \( y = \frac{1}{4}x + \frac{5}{4} \). So, the inverse function is \( g(x) = \frac{1}{4}x + \frac{5}{4} \), which corresponds to option B. Now, if you want to check your understanding of inverses, remember that for \( f \) and \( g \) to be true inverses, you should verify that \( f(g(x)) = x \) and \( g(f(x)) = x \). This means if you plug one function into the other, you should end up with your original input each time, showcasing the beautiful symmetry of inverse functions! Also, don't forget that finding inverses can get tricky with multi-variable functions or non-bijective cases! Always ensure the function is one-to-one (each output is linked to only one input) before determining an inverse, as this condition is key for an inverse to exist. Keep practicing, and you'll be a pro at finding inverses in no time!