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y=f(x)

The function. has no inverse. f(x1)=f(x2)=f(x3)=y, so. y. y=f(x). y. x. x1. x2. x3. has no inverse. But if we treat g as a function. then it does have an inverse. y. y=g(x). y. x. y. y=f(x). x. Graphs of a function and its inverse are reflections in the line y=x. y.

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y=f(x)

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  1. The function has no inverse f(x1)=f(x2)=f(x3)=y, so y y=f(x) y x x1 x2 x3

  2. has no inverse But if we treat g as a function then it does have an inverse y y=g(x) y x

  3. y y=f(x) x Graphs of a function and its inverse are reflections in the line y=x

  4. y y=f(x) y x x Slope of red tangent line is dy/dx

  5. x x y y Slope of green tangent line is dx/dy

  6. y x x y These tangents are reflections of each other in y=x and so their slopes r and s satisfy r=1/s. So dx/dy = 1/(dy/dx)

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