Inverse function by reflection

Parametric equations

If $f$ has an inverse $f^{-1}$, then

On the plane, $y=f(x)$ and $y=f^{-1}(x)$ are symmetric across the line $y=x$. If $(a,b)$ lies on $y=f(x)$, then $(b,a)$ lies on $y=f^{-1}(x)$.

Interaction

• linear / quad+ / quad / exp: switch a line, a quadratic restricted to one side of its vertex, the same quadratic with no restriction, and an exponential
• Base q: in the exponential mode, change the base of $q^x$ and watch the curve and its reflection
• Input x: drag the input, or drag P on the figure, and watch $P=(a,b)$, the reflected point $P^{\prime }=(b,a)$, and the line $y=x$
• guide: on / guide: off: show the segment joining $P$ and $P^{\prime }$ and the projections of $P$ onto the axes, marking that the input and the output have swapped
• Horizontal line test: on quad, a horizontal line counts how many times it meets the graph

What to notice

• An inverse swaps the roles of the input and the output. The graph is the reflection across $y=x$, not a new unrelated curve.
• $y=q^x$ and $y={\log }_{q}x$ are inverses, so the two graphs are symmetric across $y=x$.
• Not every function has an inverse. After the domain is restricted, the graph can pass the horizontal line test.

Related

• Exponentials and logarithms
• Exponential growth and decay
• Geometric definition of the natural logarithm
• Logarithmic scale

Further reading

• Inverse function (Wikipedia)
• Horizontal line test (Wikipedia)