Problem 11 - Precalculus with Limits

Question

In Exercises 7-14, find the inverse function of \(f\) informally. Verify that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). \(f(x) = 3x + 1\)

Step-by-Step Solution

Answer

The inverse function of \(f(x) = 3x + 1\) is \(f^{-1}(x) = (x - 1) / 3\). The properties \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) are both verified.

Step 1: Find the inverse function

To find the inverse of a function, we first rewrite the function y = f(x) as y = 3x + 1. Then we switch the roles of y and x to find the inverse: x = 3y +1. To solve for y, subtract 1 from both sides and then divide by 3 to get \(y = f^{-1}(x) = (x - 1)/3\).

Step 2: Verify that \(f(f^{-1}(x)) = x\)

Now, substitute \(f^{-1}(x)\) into function f as follows: \(f(f^{-1}(x)) = f((x - 1)/3)\). Plugging \(f(x) = f(y) = 3y + 1\) back in, we get \(f(f^{-1}(x)) = 3 * ((x - 1) / 3) + 1\). Simplifying the equation gives \(f(f^{-1}(x)) = x\)

Step 3: Verify that \(f^{-1}(f(x)) = x\)

Substitute f(x) into function \(f^{-1}\) as follows: \(f^{-1}(f(x)) = f^{-1}(3x + 1)\). Plugging \(f^{-1}(y) = (y - 1) / 3\) back in, we get \(f^{-1}(f(x)) = ((3x + 1) - 1) / 3\). Simplifying the equation gives \(f^{-1}(f(x)) = x\)

Key Concepts

Verifying Inverse Functions

When dealing with inverse functions, one of the crucial aspects to understand is the ability to verify them. Verification ensures that the inverse function indeed reverses the effect of the original function. To do this, we perform specific checks: we substitute f^-1(x) into the original function f and vice versa, checking if the compositions f(f^-1(x)) and f^-1(f(x)) both simplify to x.

It's like confirming if a lock and key are a true pair by seeing if the key can lock and unlock seamlessly. In essence, if both compositions yield the identity function, which does nothing but return the input value x, we can confidently say the functions are true inverses of each other. An easy-to-remember tagline for this concept might be: 'Inverses undo each other; if you're left with x, you've got it right!' This makes the concept of verification less daunting and more relatable for students.

Finding Inverses Algebraically

Learning to find the inverse of a function algebraically is a bit like learning a dance, you must follow specific steps to ensure success. To dance through algebra and find an inverse, start by replacing f(x) with y, thus creating an equation y = f(x). Then, perform the critical step of flipping the roles of y and x. This switch will set you on track to solving the equation for the new y, which represents f^-1(x).

After solving for y, congratulate yourself; you’ve found f^-1(x)! As with learning any dance, practice makes perfect. Over time, students will become more fluid with these steps, finding inverses with the grace of an algebraic dancer.

Function Composition

Understanding function composition is pivotal for grasping the interaction between functions, akin to understanding how ingredients combine to make a dish. Composing functions involves taking the output of one function and using it as the input for another. The notation f∘g signifies that function g is applied first, and its output is then passed to function f.

Students can think of it like this: imagine functions as machines in a factory. The output of one machine (function g) becomes the input for the next machine (function f). This process is crucial for verifying inverse functions; if two functions are indeed inverses, the composition of one with the other will result in the identity function, leaving us back with our original input.