Question

In Exercises \(13-24,\) find \(f^{-1}\) and verify that $$\left(f \circ f^{-1}\right)(x)=\left(f^{-1} \circ f\right)(x)=x$$ \(f(x)=\frac{2 x+1}{x+3}\)

Short answer

The inverse function of \(f(x)=\frac{2x+1}{x+3}\) is \(f^{-1}(x)=\frac{3x-1}{2-x}\). Both identities \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\) have been verified.

Step-by-step solution

Expression of Given Function

Express the given function \(f(x)\) in terms of \(y\), i.e., \(y=\frac{2x+1}{x+3}\)

Interchange of Variables

Exchange \(x\) and \(y\) to initiate finding the inverse function. This results in \(x=\frac{2y+1}{y+3}\)

Solving for y

Rearrange the equation from step 2, which involves cross multiplying and isolating \(y\). The solution is \(y=\frac{3x-1}{2-x}\). This is the inverse of the original function, hence \(f^{-1}(x)=\frac{3x-1}{2-x}\)

Verification of \(\left(f \circ f^{-1}\right)(x)=x\)

Confirm the first identity by substituting \(f^{-1}(x)\) into \(f(x)\). Upon simplification, the expression results in \(x\). This verifies the identity \(\left(f \circ f^{-1}\right)(x)=x\)

Verification of \(\left(f^{-1} \circ f\right)(x)=x\)

Confirm the second identity by substituting \(f(x)\) into \(f^{-1}(x)\). The outcome, after simplification, should equate to \(x\), demonstrating the verification of \(\left(f^{-1} \circ f\right)(x)=x\)