Temperature as function of carbon dioxide The carbon stored in fossil fuels must have been taken out of the earth atmosphere from CO2 in past periods by plants. Thereby reducing the amount of CO2 drastically and increasing the amount of O2 to current levels. But this change in the composition of the atmosphere did not cause an enormous cooling down of the planet. So how can it be that a small increase of CO2 now would significantly increase temperature. These processes should be proportional to the percentage of CO2?
 A: It's worth noting that the amount of 'available' $\mathrm{CO_2}$ in the system is not constant over very long geological times.  $\mathrm{CO_2}$ gets dumped into the atmosphere by vulcanism & other geological processes, and this then gets scavenged from it by life.  So, although all the $\mathrm{CO_2}$ now stashed as coal & oil came from the atmosphere, there was no point at which the atmosphere had all the $\mathrm{CO_2}$ that is currently stashed in biomass & its products in it, because other processes are adding $\mathrm{CO_2}$ to the system over time.
That being said I think that it is indeed the case that before life started scavenging $\mathrm{CO_2}$ from the atmosphere it was indeed a very major component of it, and there was a really pretty large greenhouse effect.  That's a good thing as the Sun was significantly dimmer, and that greenhouse effect was enough to keep the planet warm enough to stop the oceans freezing, which would have at least severely hindered the origin of life.
It's quite easy to see how large the effect must have been in fact.  The Sun has got about $30\%$ more luminous over its lifetime: we know it now has a power of about $3.9\times 10^{26}\,\mathrm{W}$, so in the early Solar System its power output was about $3\times 10^{26}\,\mathrm{W}$.  And you can then work out how hot a simple-minded black-body Earth would be in both cases (so, ignoring any greeenhouse effect or albedo, and also ignoring changes in orbital radius -- I think changes in orbital radius have been fairly small although I may be wrong).
Today, such a black-body Earth would have a temperature of about $279\,\mathrm{K}$, or about $6\,\mathrm{{}^\circ C}$.  This is about $7\,\mathrm{K}$ below the actual average surface temperature.
At the start of the solar system, a black-body Earth would have a temperature of about $261\,\mathrm{K}$ or about $-11\,\mathrm{{}^\circ C}$.
So without a really significant greenhouse effect the oceans would have frozen and life probably would not have got started.
It's also worth noting that relatively large changes in surface temperature have indeed happened over the life of the Earth: life is perfectly possible with a surface temperature significantly warmer than it is now, and significantly colder (I'm not going to quote figures because I don't know them but probably ten degrees in either direction is perfectly fine, fifty isn't).  When temperature changes very rapidly with time, as we are now experiencing, it's not good news, but even then it's not actually a threat to the existence of life as a whole.
An example of what happens when there is no life to scavenge $\mathrm{CO_2}$ dumped into the atmosphere over long periods by geological processes is Venus, of course.
A: The relationship is nonlinear, not even logarithmic.  To oversimplify a very complicated subject ...  
The key issue is the altitude of the tropopause, where convective transport through the troposphere gives way to radiative transport through the stratosphere.  Convective instability ends where the adiabatic temperature gradient (a.k.a. lapse rate) exceeds the temperature gradient needed to drive radiative transport through blocked spectral bands, the latter being proportional to the opacity of the air in the blocked bands.  (A wavelength is considered blocked from a given altitude if the air above is optically thick.)  Increasing the opacity would raise altitude of the tropopause.  Given the exponential thinning of the stratosphere, doubling the opacity would raise the tropopause about 4.5 km.  Each extra kilometer of altitude could raise the surface temperature as much as 6.5 degC, according to the effective lapse rate of the Standard Atmosphere, and the theoretical adiabatic lapse rate in dry air is even higher, 9.8 degC/km.
Doubling CO2 would not even double the opacity of the dry air in the stratosphere, since CO2 only blocks a band from ~13 to ~17 microns.  Moreover, much of this band is said to be “saturated” since absorption cannot exceed 100%, no matter how much CO2 there might be.    Wikipedia has a nice figure of the stratospheric transmission spectra, which shows that additional CO2 would merely broaden the shoulders of the absorption band.  
A: I got the following conversion rate [(CO2/carbon), by weight] from the following page [which is not an endorsement of the (main) point of the page].
https://skepticalscience.com/print.php?r=45
At 400ppm, there is (7.8 * 400 =) 3120Gt of CO2 in the atmosphere (the latter being 5148000Gt, from the following).
Is the total mass of Earth's atmosphere essentially constant in time?
Annually, the CO2 is exchanged at the rate of about 750Gt in and 750Gt out.  [I am currently trying to figure out why there are two different numbers, 215 being the other one.  My current best guess is that one is carbon, and the other is CO2 (= carbon * 44/12)].  This is the combined effect of the various players (of which fossilizing is one) in “the Carbon [Dioxide?] Cycle”.  Some arbitrary references follow.
[790]  https://skepticalscience.com/human-co2-smaller-than-natural-emissions.htm
[775]  https://cdiac.ess-dive.lbl.gov/pns/graphics/c_cycle.htm
[750]  https://www.globe.gov/documents/355050/14396119/IntroductionToGlobalCarbonCycle.pdf/650d0e25-3944-4ac3-959d-e8da347653f5
[750, 216]  https://wattsupwiththat.com/2021/06/07/carbon-cycle/
[795]  https://wattsupwiththat.com/2017/05/21/examining-the-carbon-dioxide-cycle/
[215]  https://wattsupwiththat.com/2019/05/01/a-story-of-co2-data-manipulation/
[Offhand… human emissions have climbed to around 40Gt/year.]
I was going to cite an article that says that the “half-life” of CO2 in the atmosphere is 10 years… but I can not find it, and I found another article that said 40 years while I was looking (and I saw 1.5% remaining after 100 years).  Simple maths says that… if 750/3120 of the CO2 is removed in any given year [and the same added], that is 24% exchanged annually — leaving 76% of what was there remaining.  The former seems a bit high; it leaves just over 25% after 5 years, 11% after 8 years, and 6.4% after 10 years.  (775 would be similar, reaching 1% after 16 years, vs 16.75 for 750.)
Anyway, the point is that the amount of CO2 in the atmosphere is kept static (or changed gradually) by a balance of huge inflows and huge outflows (of which (again) fossilizing is only one).
————
As for the title (“temperature as a function of CO2”)…
Another part of the picture is that CO2 is a far less significant greenhouse gas than is… water vapour (and there are a few other bit players as well (and methane fails to qualify)).
Finally, we note that the effect of CO2 on the Earth’s temperature is logarithmic — to get a linear increase in temperature, you have to repeatedly double the amount of CO2… and the effect is largely exhausted at the current levels.
The CAGW [alarmist] position is that the system is driven by {the increase in CO2} driving an increase in temperature, which drives an increase in water vapour, which drives a greater increase in temperature.
Thus, the “denier” position is that human emissions of CO2 are not driving the temperature, and the true “alarmist” position is that the temperature is more a function of water vapour.
I should note that there are good reasons to question the CAGW tenet (that temperature is a function of CO2) — over and above the fact that CAGW itself does not hold that.  Unfortunately, the level of screaming propaganda has reached the point that people actively refuse to even consider the possibility that it might be wrong… so I am not going to bother.
