Question

How do you find $\left( goh \right)\left( 1 \right)$ given $g\left( x \right)={{x}^{2}}+4+2x$ and $h\left( x \right)=-3x+2$?

Answer 1

As we know that if we have two functions like $f(x)$ and $g(x)$ then we can write the composite function as $\left( fog \right)x=f\left( g\left( x \right) \right)$. So by using this concept we will find the composite of given functions. Then substitute the value 1 in place of x to get the desired answer.

Complete step by step solution:
We have been given two functions $g\left( x \right)={{x}^{2}}+4+2x$ and $h\left( x \right)=-3x+2$.
We have to find the value of $\left( goh \right)\left( 1 \right)$.
Now, we have to find the composite function for the given functions. We know that the general form of a composite function will be $\left( goh \right)\left( x \right)$.
Now, we know that $\left( goh \right)x=g\left( h\left( x \right) \right)$
Now, substituting the values we will get
$\Rightarrow \left( goh \right)x=g\left( -3x+2 \right)$
Now, again substituting the value of x in the above obtained equation from the given function $g(x)$. Then we will get
$\Rightarrow \left( goh \right)x={{\left( -3x+2 \right)}^{2}}+4+2\left( -3x+2 \right)$
Now, simplifying the above obtained equation we will get
$\begin{aligned} & \Rightarrow \left( goh \right)x={{\left( -3x \right)}^{2}}+{{2}^{2}}+2\times 2\times \left( -3x \right)+4-6x+4 \\\ & \Rightarrow \left( goh \right)x=9{{x}^{2}}+4-12x+4-6x+4 \\\ & \Rightarrow \left( goh \right)x=9{{x}^{2}}-18x+12 \\\ \end{aligned}$
Now, substituting $x=1$ in the above obtained equation we will get
$\Rightarrow \left( goh \right)\left( 1 \right)=9\times {{1}^{2}}-18\times 1+12$
Now, simplifying the above obtained equation we will get
$\begin{aligned} & \Rightarrow \left( goh \right)\left( 1 \right)=9-18+12 \\\ & \Rightarrow \left( goh \right)\left( 1 \right)=3 \\\ \end{aligned}$
Hence we get the value of $\left( goh \right)\left( 1 \right)=3$

Note: A composite function is formed when one function is substituted into another function. The point to be remembered is that if the given functions are restricted to some interval for which we have to find the composite function then first we need to check the ranges and domains of both the functions.