Graph of $\frac 1x$ and what is an inverse relationship?

Question

I understand that the graph of $\frac 1x$ is not a line as it does not agree with the form $y=mx+c$ and the properties of a linear relationship. However I'm struggling intuitively to understand this. When we graph $y=x$, we find as $x$ doubles $y$ also doubles and so on. They have a proportional relationship. As $x$ doubles in $\frac 1x$, the $y$ values half and so on. This is an indirectly proportional relationship. My intuition makes me think that it should also be a line but sloped the other way. However this is not the case. Why? I think my problem lies in intuitively understanding why the graph of $\frac 1x$ increases exponentially. As when $x$ doubles $y$ halves it doesn't make sense to me why the curve increases so rapidly.

Answer 1

The function $f(x):=1/x$ does not change exponentially. An exponential function would be of the form $g(x):=ab^x$ for $b>0$.

You have already identified that as $x$ doubles, $y$ halves. Alternatively, you could say as $x$ halves, $y$ doubles; this observation alone should give you a sense why $f$ spikes so rapidly near the origin.

You ask why $f$ does not look like a line. Note that a linear function has a constant rate of change, so in essence, you are asking why $f$ doesn't have a constant rate of change.

Note that $f$ changes much more at smaller values of $|x|$. For instance, $$f(1)-f(2)=1-1/2=1/2>1/6=1/2-1/3=f(2)-f(3).$$ In words, this says if I gave you a cake and took half of it away from you, you would be left with more cake than if I gave you half the cake and took a third of the cake away from you.

Answer 2

In the precursor Question, I suggested that you may be unclear about inverse proportionality $(xy=7)$ versus direct proportionality $(y=7x)$.

In this Question, though, you appear to be conflating

• inverse proportionality $(xy=7)$

  tripling $x$ correspondingly divides $y$ by three

• negative direct proportionality $(y=-7x)$

  tripling $x$ correspondingly triples $y;$ $x$ and $y$ have opposite signs

• exponential decrease $(y=7e^{-5x})$

  $y$ is undergoing constant percentage decay.

For $x>0:$

A key difference between positive inverse proportionality $(xy=7)$ and a negative linear relationship $(y=-5x+7)$ is that in the former, $x$ and $y$ have the same sign, so as $x$ gets large, $y$ doesn't get to decrease beyond the $x$-axis (its floor), so its continued decrease significantly slows down.