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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?

enter image description here

gt6989b
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traggatmot
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3 Answers3

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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 ☺

John Molokach
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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

Leonidas Lanier
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    From equation (2) I would just multiply by 25 and then raise both sides to $1/1.7$. No logarithms needed. (I am not sure the OP wanted the answer...) – John Molokach Jul 11 '16 at 02:22
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    The logarithmic property isn't needed though, right? After dividing each side by 0.04, can't we simply raise each side to the power of 10/7? Nah, didn't want the answer - just couldn't for the life of me remember something I clearly learned in high school 17 years ago....oh the brain... – traggatmot Jul 11 '16 at 02:22
  • Yeah that'd be easier actually – Leonidas Lanier Jul 11 '16 at 02:22
  • @traggatmot did you mean 10/17 instead of 10/7? Also, do you want to accept an answer so we can close this question out? – John Molokach Jul 11 '16 at 17:46
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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.

Narasimham
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