Two functions f and g are defined on the set R of real numbers by \(f : x \to 2x - 1\) and \(g : x \to x^{2} + 1\). Find the value of \(f^{-1} \circ g(3)\).
Question 1 Report
Two functions f and g are defined on the set R of real numbers by \(f : x \to 2x - 1\) and \(g : x \to x^{2} + 1\). Find the value of \(f^{-1} \circ g(3)\).
Answer Details
We first need to find the composition \(f^{-1} \circ g(x)\), which means we need to find the inverse function of f, denoted by \(f^{-1}(x)\).
To find \(f^{-1}(x)\), we solve the equation \(y = 2x - 1\) for x:
\(y + 1 = 2x\)
\(x = \frac{y + 1}{2}\)
Thus, \(f^{-1}(x) = \frac{x + 1}{2}\).
Now, we can find \(f^{-1} \circ g(3)\) by first computing g(3):
\(g(3) = 3^{2} + 1 = 10\)
Then, we can plug g(3) into the composition \(f^{-1} \circ g(x)\):
\((f^{-1} \circ g)(3) = f^{-1}(g(3)) = f^{-1}(10) = \frac{10 + 1}{2} = \frac{11}{2}\)
Therefore, the value of \(f^{-1} \circ g(3)\) is \(\frac{11}{2}\), which is the correct answer.
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