Question

How do you find (f of g of h) if $f(x)={{x}^{2}}+1$, $g(x)=2x$ and $h(x)=x-1$.

Answer 1

Hint: In the above question you were asked to find (f of g of h), it is given that $f(x)={{x}^{2}}+1$, $g(x)=2x$ and $h(x)=x-1$. This is a problem of composition function and (f of g of h) is the composite that composes f with g with h. So let us see how we can solve this problem.

Complete Step by Step Solution:
In the given question we have to find (f of g of h) that is (f of g of h).
It is given that:
 1. $f(x)={{x}^{2}}+1$
 2. $g(x)=2x$
 3. $h(x)=x-1$
The verbal description of the above three composite functions is stated below:
1. f takes the square of the number and then adds 1
2. g doubles the number
3. h subtracts 1 from the number
So, the description of the composite function (f of g of h) in the sequence is:
1. Subtract 1
2. Double
3. Square
4. Add 1
So, the process of the above symbol is:
 $x\to x-1\to 2(x-1)\to {{(2(x-1))}^{2}}\to {{(2(x-1))}^{2}}+1$
So,
 $\Rightarrow \left( f\text{ }of\text{ }g\text{ }of\text{ }h \right)(x)=f(g(h(x)))$
 $={{(2(x-1))}^{2}}+1$
 $=4({{x}^{2}}-2x+1)+1$
After multiplying 4 with $({{x}^{2}}-2x+1)$ we get,
 $=4{{x}^{2}}-8x+4+1$
 $=4{{x}^{2}}-8x+5$

Therefore, (f of g of h) is $4{{x}^{2}}-8x+5$

Note:
In the above solution, we solved the problem with the composite function. The (f of g of h) means the product of f, g and h. We also need to understand the verbal description of these functions. In our problem was, subtract 1, double, square, and then add 1.