How do nuclear fusion and fission both release energy when they are the opposite process?

fusion releases energy that is left over after the process the 2 atoms combine but do not use up everything in the process so all the extra is released from 2 atoms so it's a higher energy output then fision which releases the energy that is inside the single atom

The energy available from changing the composition of a nucleus
is generally 'up' as you move toward iron, the most stable of the nuclei.
You get energy as fou fuse up in weight toward iron from lighter elements
and down in weight from the heavier ones.
We fuse hydrogen, we split uranium and plutonium.

There is less matter present after both processes. The difference between the mass before and the mass after is converted to energy according to

E= Mc^2

Short answer: Nuclear fission (splitting) releases energy when it happens to heavy nuclei (like uranium and plutonium) but absorbs energy when it happens to light nuclei (like helium or carbon).

Nuclear fusion (joining) releases energy when it happens to light nuclei but absorbs energy when it happens to heavy nuclei.

The cut-off point is iron.  Any element significantly lighter than iron will release energy when it fuses, and any element significantly heavier than iron will release energy when it fissions.

Long Answer:

A proton and a neutron both have a rest mass of about 1 AMU (atomic mass unit).  Specifically, the rest mass of a neutron is 1.008665 AMU and the rest mass of a proton is slightly less at 1.007276 AMU.  Protons and neutrons join together to form atomic nuclei.  Now you might think that the mass of an atomic nucleus is simply equal to the sum of its parts, but that's not how it works.  For example, let's consider the nucleus of helium-4, which has 2 protons and 2 neutrons.  Based on the masses of protons and neutrons, you'd expect its mass to be:

(2 x 1.008665 AMU) + (2 x 1.007276 AMU) = 4.031882 AMU

The ACTUAL mass of a He-4 nucleus is 4.00153 AMU

Which is a difference of 0.03035 AMU.  The difference between what the nucleus SHOULD weigh if you added up the separate masses of its nucleons and what it ACTUALLY weighs is called "mass defect".  As it turns out, EVERY nucleus has a mass defect, meaning every nucleus weighs less than you'd expect it to.

So where does that missing mass go?  Well, you've probably heard of Einstein's famous mass-energy equivalence formula: E = mc².  This equation indicates that mass and energy are two faces of the same coin and can be converted back and forth.  As it turns out, though, a little bit of mass produces a LOT of energy.  The mass defect of a nucleus is actually converted into energy when the nucleus is formed from its parts.  This energy is known as "binding energy".  The binding energy is also the amount of energy you'd need to put into an atomic nucleus to separate it completely into protons and neutrons.  If you applied the binding energy to an atomic nucleus, it would be converted into mass.  The separate protons and neutrons that resulted would be collectively heavier than the nucleus from which they came.

If you convert helium-4's mass defect into binding energy, you come up with 28.3 MeV (Mega electron-volts, a unit of energy often used to describe atomic phenomena).  Since helium-4 is made of four nucleons (2 protons + 2 electrons), that's about 7.08 MeV per nucleon.

Three helium-4 nuclei can undergo fusion to form carbon-12.  The binding energy of the C-12 nucleus is 92.2 MeV, or about 7.68 MeV per nucleon.  In other words, the fusion of three He-4 nuclei into a single C-12 nucleus causes a release of about 0.60 MeV per nucleon involved.

The highest binding energy per nucleon occurs in a few isotopes of nickel and iron.  If you fuse any two nuclei to form something lighter than iron, there will be a net release of binding energy, meaning the process will be exothermic.  Fissioning an isotope lighter than iron will result in a net DECREASE in binding energy (meaning the nucleus must absorb energy).  Therefore, the fission of any light isotope is endothermic.

When you consider nuclei heavier than iron, the situation reverses.  As I said before, the maximum binding energy per nucleon occurs in isotopes of nickel and iron.  In heavier isotopes the binding energy per nucleon decreases.  To fuse two nuclei heavier than iron would result in a net DECREASE in binding energy per nucleon, meaning it would have to absorb energy (endothermic).  However, fissioning heavy nuclei results in a net INCREASE in binding energy per nucleon, meaning fission is an exothermic process for heavy nuclei.

So yes, fission and fusion ARE opposite processes, but fission is exothermic when it happens to heavy nuclei and fusion is exothermic when it happens to light nuclei.

I hope that helps.  Good luck!

Yep, binding energy.  What Lucas said.  Look it up.  It's cool.