Question

let (g(x)) be the inverse of (f(x)=x^{3}+2x + 2). without finding a formula for (g(x)), calculate (g(5)). then calculate (g(5)).

(g(5)=)

(g(5)=)

Explanation:

Step1: Recall inverse - function property

If \(g(x)\) is the inverse of \(f(x)\), then \(f(g(x))=x\) and \(g(f(x)) = x\). We want to find \(g(5)\), so we need to find \(x\) such that \(f(x)=5\). That is, we solve the equation \(x^{3}+2x + 2=5\), or \(x^{3}+2x - 3=0\).

Step2: Factor the equation

We can factor \(x^{3}+2x - 3\) as \((x - 1)(x^{2}+x + 3)=0\). The quadratic factor \(x^{2}+x + 3\) has discriminant \(\Delta=1^{2}-4\times3=- 11<0\), so its roots are complex. The real - root of \(x^{3}+2x - 3 = 0\) is \(x = 1\). So \(g(5)=1\).

Step3: Recall the formula for the derivative of an inverse function

The formula for the derivative of an inverse function is \(g^{\prime}(y)=\frac{1}{f^{\prime}(g(y))}\). We want to find \(g^{\prime}(5)\), so we first find the derivative of \(f(x)\). Using the power rule, \(f^{\prime}(x)=3x^{2}+2\).

Step4: Evaluate \(f^{\prime}(g(5))\)

Since \(g(5)=1\), we evaluate \(f^{\prime}(1)\). Substitute \(x = 1\) into \(f^{\prime}(x)\): \(f^{\prime}(1)=3\times1^{2}+2=5\).

Step5: Calculate \(g^{\prime}(5)\)

Using the formula \(g^{\prime}(5)=\frac{1}{f^{\prime}(g(5))}\), and since \(f^{\prime}(g(5))=f^{\prime}(1)=5\), we have \(g^{\prime}(5)=\frac{1}{5}\).

Answer:

\(g(5)=1\)
\(g^{\prime}(5)=\frac{1}{5}\)