Question

In Exercises \(13-24,\) find \(f^{-1}\) and verify that $$\left(f \circ f^{-1}\right)(x)=\left(f^{-1} \circ f\right)(x)=x$$ \(f(x)=5-4 x\)

Short answer

The inverse function of \(f(x) = 5 - 4x\) is \(f^{-1}(x) = \frac{5 - x}{4}\). The Given conditions \(\left(f \circ f^{-1}\right)(x)=\left(f^{-1} \circ f\right)(x)=x\) are satisfied, so our found inverse function is correct.

Step-by-step solution

First step: Find the inverse

To find the inverse of a function, we need to switch the roles of \(x\) and \(y\). Here, we'll replace the \(f(x)\) with a \(y\), swap \(x\) and \(y\), and then solve for \(y\). Let \(y = 5 - 4x\). Now, interchange \(x\) and \(y\); we get \(x = 5 - 4y\). Solving this for \(y\), we find that \(y = \frac{5 - x}{4}\) . So, \(f^{-1}(x) = \frac{5 - x}{4}\).

Second step: Verify the first property

We need to prove that \(f(f^{-1}(x)) = x\). Substituting \(f^{-1}(x)\) into \(f(x)\), we get \(f(f^{-1}(x)) = 5 - 4 \cdot \frac{5 - x}{4}\). Simplifying this results in \(x\). This verifies the first property.

Third step: Verify the second property

We need to prove that \(f^{-1}(f(x)) = x\). Substituting \(f(x)\) into \(f^{-1}(x)\), we get \(f^{-1}(f(x)) = \frac{5 - (5 - 4x)}{4} = x\). This verifies the second property.