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If $g(x) = {e^{2x}} + {e^x} - 1$ and $h(x) = 3{x^2} - 1,$ the value of $g(h(0))$ is

Last updated date: 13th Jun 2024
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Hint: Here first we will take value $h(0)$ and here take $x$ value as zero then we will substitute the $x$ value as zero in $h(x)$ . Here we will get the answer for $h(x)$ . Then we will substitute the answer of $h(x)$ in $g(x)$. Finally, we will get the answer for this question.

Complete step-by-step solution:
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}$

So finally, we will get the answer for this question as $ \dfrac{1}{e}$

Note: The reciprocal of a number is $1$ divided by the number. The reciprocal of a number is also called its multiplicative inverse. The product of a number and its reciprocal is $1$ . The reciprocal of a fraction is found by flipping its numerator and denominator. This reciprocal is mainly used for changing the minus value.
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.