what’s the inverse function

Accepted Solution

Answer:[tex]f^{-1}(x)=-2x+6[/tex].Step-by-step explanation:[tex]y=f(x)[/tex][tex]y=3-\frac{1}{2}x[/tex]The biggest thing about finding the inverse is swapping x and y.  The inverse comes from switching all the points on the graph of the original.  So a point (x,y) on the original becomes (y,x) on the original's inverse.Sway x and y in:[tex]y=3-\frac{1}{2}x[/tex][tex]x=3-\frac{1}{2}y[/tex]Now we want to remake y the subject (that is solve for y):Subtract 3 on both sides:[tex]x-3=-\frac{1}{2}y[/tex]Multiply both sides by -2:[tex]-2(x-3)=y[/tex]We could leave as this or we could distribute:[tex]-2x+6=y[/tex]The inverse equations is [tex]y=-2x+6[/tex].Now some people rename this [tex]f^{-1}[/tex] or just call it another name like [tex]g[/tex].[tex]f^{-1}(x)=-2x+6[/tex].Let's verify this is the inverse.If they are inverses then you will have that:[tex]f(f^{-1}(x))=x \text{ and } f^{-1}(f(x))=x[/tex]Let's try the first:[tex]f(f^{-1}(x))[/tex][tex]f(-2x+6)[/tex]  (Replace inverse f with -2x+6 since we had [tex]f^{-1})(x)=-2x+6[/tex][tex]3-\frac{1}{2}(-2x+6)[/tex]  (Replace old output, x, in f with new input, -2x+6)[tex]3+x-3[/tex]  (I distributed)[tex]x[/tex]Bingo!Let's try the other way.[tex]f^{-1}(f(x))[/tex][tex]f^{-1}(3-\frac{1}{2}x)[/tex] (Replace f(x) with 3-(1/2)x since [tex]f(x)=3-\frac{1}{2}x[/tex])[tex]-2(3-\frac{1}{2}x)+6[/tex] (Replace old input, x, in -2x+6 with 3-(1/2)x since [tex]f(x)=3-\frac{1}{2}x[/tex])[tex]-6+x+6[/tex] (I distributed)[tex]x[/tex]So both ways we got x.We have confirmed what we found is the inverse of the original function.