In my precalculus textbook there is a question that reads, \"Find f(1/2) if (f(g

Question

In my precalculus textbook there is a question that reads, "Find f(1/2) if (f(g(x)) = (x^4 + x^2)/(1+x^2) and g(x)= 1- x^2. I couldn't figure out how to solve this problem so I looked at the answer key, which showed:

(f o g)(x) = f(g(x))
= f(1-x^2)
= x^2(x^2+1)/(1+ x^2)
= x^2
= -(1-x^2) + 1

So, f(x) = -x + 1 and f(1/2)= -1/2+1 = 1/2.

I don't know how they got -(1-x^2) + 1 and how that translates to -x +1. I would really like to understand the process of getting the answer instead of just knowing the answer itself. I would appreciate it if anyone could help me understand.

Explanation / Answer

Let u=1?x2?f(g(x))=f(u). Simple change of variables. f(u)=f(g(x))=x4+x21+x2=x2(x2+1)1+x2=x2(x2+1)x2+1=x2. Just mechanics. But I need to express f(u) in terms of u. u=1?x2?x2=1?u. Any problem there? So f(u)=1?u?f(x)=1?x. Thus f(12)=1?12=12. Now what your book is doing is a shortcut. It gets to f(g(x))=x2. But g(x)=1?x2. How do I translate x2 into something that is expressed in terms of 1?x2. First step is that I need a negative. Simple x2=?(?x2). You buy that? Second step is that I need a + 1. Well if I add a 1 and simultaneously subtract a 1 that is zero, and I can add zero without changing anything. x2=?(?x2)=0?(?x2)=1?1?(?x2)=1?(1?x2). This trick of putting something into an expression by adding it and subtracting it at the same time is a very common trick.

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