Determine equation for reflection of equation around y=x line

Question

In the graph below I have two lines plotted:

1. $y = x$ (solid line)
2. $y = 0.04 x^{1.7}$ (hollow dots)

How can I come up with an equation that is a mirror image (if that's the terminology) of the $y = 0.04 x^{1.7}$ equation? So that I have two equations that are symmetric about the $y=x$ line?

Answer 1

Let's call $f(x)=0.04x^{1.7}$. The function is one-to-one and therefore has an inverse. To find the equation of the inverse, solve the equation $$x=0.04y^{1.7}$$ for $y$. I will leave that part up to you ☺

Answer 2

We have this:

$$y=0.04x^{1.7}$$

Exchanging $x$s and $y$s:

$$x=0.04y^{1.7}$$

Multiplying each side by $25$ and using logarithms:

$$\ln25x=1.7\ln y$$

Dividing by $1.7$ and simplifying:

$$e^{\ln25x/1.7}=y$$

Using this logarithm property:

$$(25x)^\frac{1}{1.7}=y$$

And we're finished

Answer 3

To find the equation of f(x,y)=0 after reflection about $x=y$ we just swap $x,y$ positions to get f(y,x) =0 in the same relative arrangement/setting of the equation variables.