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In Question given that

$

g(x) = {e^{2x}} + {e^x} - 1 \\

h(x) = 3{x^2} - 1

$

Now we will take $x$ as zero in given question

$g(h(0)) = g(3{x^2} - 1)$

So here $x = 0$

$

= g(3(0) - 1) \\

= g(0 - 1) \\

= g( - 1)

$

Using above answer again we take $x$ as $ - 1$

Substitute the value $x$ as $ - 1$ in $g(x)$ equation

$g(x) = {e^{2x}} + {e^x} - 1$

$ = {e^{2( - 1)}} + {e^{ - 1}} - 1$

If we remove minus value, we will take reciprocal for these values

$ = \dfrac{1}{{{e^2}}} + \dfrac{1}{e} - 1$

After taking lcm for above equation we will get

$ = \dfrac{{1 + e - 1}}{{{e^2}}}$

Here we will remove $ + 1$ and the $ - 1$ we will get the answer

$ = \dfrac{e}{{{e^2}}}$

Here numerator $e$ and denominator $e$ will be cancelled we will get the answer

$ = \dfrac{1}{e}$

Here we will be using the concept relation and function. A function is a relation which describes that there should be only one output for each input we can say that a special kind of relation (a set of ordered pairs), which follows a rule. Every $x$ value should be associated with only one $y$ value is called a function.