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A relation R:A→AR:A→A is said to be reflexive if, for every element aa in the set AA (where AA is a non-empty set), the ordered pair (a,a)(a,a) belongs to the relation RR. In simpler terms, reflexive relations include every element paired with itself in the set.

To prove that the function f:R→Rf:R→R given by f(x)=2xf(x)=2x is one-to-one (injective), we need to show that if f(x1)=f(x2)f(x1)=f(x2), then x1=x2x1=x2 for all x1,x2x1,x2 in the domain.

Let's assume f(x1)=f(x2)f(x1)=f(x2): 2x1=2x22x1=2x2

Now, we'll solve for x1x1 and x2x2: x1=x2x1=x2

Since x1=x2x1=x2, it means that for any two inputs x1x1 and x2x2 that produce the same output under the function f(x)=2xf(x)=2x, those inputs must be the same. This proves that the function f(x)=2xf(x)=2x is one-to-one.

To prove that (f+g)∘h=f∘h+g∘h(f+g)∘h=f∘h+g∘h, let's start by understanding what (f+g)∘h(f+g)∘h means:

(f+g)∘h(x)=(f+g)(h(x))=f(h(x))+g(h(x))(f+g)∘h(x)=(f+g)(h(x))=f(h(x))+g(h(x))

Now, let's find (f∘h+g∘h)(x)(f∘h+g∘h)(x):

f∘h(x)+g∘h(x)=f(h(x))+g(h(x))f∘h(x)+g∘h(x)=f(h(x))+g(h(x))

This expression is identical to what we found for (f+g)∘h(x)(f+g)∘h(x). Hence, we can conclude that (f+g)∘h=f∘h+g∘h(f+g)∘h=f∘h+g∘h.

To find (g∘f)(x)(g∘f)(x), which is the composition of g(x)g(x) with f(x)f(x), we substitute f(x)f(x) into g(x)g(x) wherever we see xx.

Given:

f(x)=∣x∣f(x)=∣x∣ g(x)=∣5x+1∣g(x)=∣5x+1∣

We first find f(x)f(x):

f(x)=∣x∣f(x)=∣x∣

And then substitute it into g(x)g(x):

g(f(x))=∣5(∣x∣)+1∣g(f(x))=∣5(∣x∣)+1∣

Now, ∣x∣∣x∣ can be either xx if x≥0x≥0 or −x−x if x<0x<0.

So, ∣5(∣x∣)+1∣∣5(∣x∣)+1∣ will be:

If x≥0x≥0: g(f(x))=∣5x+1∣g(f(x))=∣5x+1∣

If x<0x<0: g(f(x))=∣−5x+1∣g(f(x))=∣−5x+1∣

Nitrogen dioxide (NO2NO2) dimerizes to form dinitrogen tetroxide (N2O4N2O4) due to the presence of unpaired electrons on each nitrogen atom in the NO2NO2 molecule. This dimerization process is a result of the tendency of molecules with unpaired electrons to pair up and form more stable configurations.

In the gas phase, NO2NO2 exists predominantly as a reddish-brown dimer, N2O4N2O4, which is colorless. The dimerization reaction can be represented as:

2NO2⇌N2O42NO2⇌N2O4

This process is reversible, meaning that N2O4N2O4 can dissociate back into NO2NO2 molecules. The equilibrium between NO2NO2 and N2O4N2O4 depends on factors such as temperature, pressure, and concentration.

The dimerization of NO2NO2 to form N2O4N2O4 is an important reaction in atmospheric chemistry. In polluted urban environments, NO2NO2 is often emitted from vehicles and industrial sources. When NO2NO2 reacts with other pollutants and undergoes dimerization to form N2O4N2O4, it can contribute to the formation of smog and other harmful atmospheric conditions.

The complex Co(NH3)5(NO2)2 exhibits two types of isomerism:

1. Coordination Isomerism: Coordination isomers occur when the ligands in a complex exchange places with anionic or neutral ligands outside the coordination sphere. In this complex, NO2 and NO3 can interchange positions, leading to the formation of coordination isomers.

2. Ionization Isomerism: Ionization isomers arise when there's a difference in the location of a ligand within a complex or between an ion and a molecule. In this case, the NO3^- ions in the coordination sphere can exchange positions with the NO3^- ions outside the coordination sphere.