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

How do you find (f of g of h) if f(x)=x2+1f(x)={{x}^{2}}+1, g(x)=2xg(x)=2x and h(x)=x−1h(x)=x-1.

Explanation

Solution

In the above question you were asked to find (f of g of h), it is given that f(x)=x2+1f(x)={{x}^{2}}+1, g(x)=2xg(x)=2x and h(x)=x−1h(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)=x2+1f(x)={{x}^{2}}+1
2. g(x)=2xg(x)=2x
3. h(x)=x−1h(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→x−1→2(x−1)→(2(x−1))2→(2(x−1))2+1x\to x-1\to 2(x-1)\to {{(2(x-1))}^{2}}\to {{(2(x-1))}^{2}}+1
So,
⇒(f of g of h)(x)=f(g(h(x)))\Rightarrow \left( f\text{ }of\text{ }g\text{ }of\text{ }h \right)(x)=f(g(h(x)))
=(2(x−1))2+1={{(2(x-1))}^{2}}+1
=4(x2−2x+1)+1=4({{x}^{2}}-2x+1)+1
After multiplying 4 with (x2−2x+1)({{x}^{2}}-2x+1) we get,
=4x2−8x+4+1=4{{x}^{2}}-8x+4+1
=4x2−8x+5=4{{x}^{2}}-8x+5

Therefore, (f of g of h) is 4x2−8x+54{{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.