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Let f(x)=(2x-2)/(x^(2)+1) and g(x)=x^(2)+1. What is the value of f(1+g(1)) ?

Let f(x)=2x−2x2+1 f(x)=\frac{2 x-2}{x^{2}+1} and g(x)=x2+1 g(x)=x^{2}+1 . What is the value of f(1+g(1)) f(1+g(1)) ?

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Full solution

Q. Let f(x)=2x−2x2+1 f(x)=\frac{2 x-2}{x^{2}+1} and g(x)=x2+1 g(x)=x^{2}+1 . What is the value of f(1+g(1)) f(1+g(1)) ?
  1. Find g(1)g(1): First, let's find the value of g(1)g(1).
    g(x)=x2+1g(x) = x^2 + 1
    g(1)=12+1g(1) = 1^2 + 1
    g(1)=1+1g(1) = 1 + 1
    g(1)=2g(1) = 2
  2. Calculate 1+g(1)1 + g(1): Now, let's calculate 1+g(1)1 + g(1).\newline1+g(1)=1+21 + g(1) = 1 + 2\newline1+g(1)=31 + g(1) = 3
  3. Substitute xx in f(x)f(x): Next, we will substitute xx in f(x)f(x) with the value we found, which is 33.
    f(x)=2x−2x2+1f(x) = \frac{2x - 2}{x^2 + 1}
    f(3)=2⋅3−232+1f(3) = \frac{2\cdot 3 - 2}{3^2 + 1}
  4. Simplify f(3)f(3): Now, let's simplify the numerator and the denominator of f(3)f(3).\newlinef(3)=6−29+1f(3) = \frac{6 - 2}{9 + 1}\newlinef(3)=410f(3) = \frac{4}{10}
  5. Reduce fraction: Finally, we simplify the fraction 410\frac{4}{10} to its simplest form.\newline410\frac{4}{10} can be reduced by dividing both the numerator and the denominator by 22.\newline410=25\frac{4}{10} = \frac{2}{5}

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