Problem 39 - Algebra for College Students

Question

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=-3 x-4 $$

Step-by-Step Solution

Answer

The inverse is \(f^{-1}(x) = -\frac{x+4}{3}\), and it verifies both composition identities.

Step 1: Set Up the Equation for Inversion

To find the inverse function, we start by setting the function equal to y: \[ y = -3x - 4 \]Our goal is to express x in terms of y.

Step 2: Solve for x

To isolate x, first add 4 to both sides: \[ y + 4 = -3x \]Next, divide both sides by -3 to solve for x: \[ x = -\frac{y + 4}{3} \]

Step 3: Swap x and y

To express this equation as the inverse function, swap x and y:\[ y = -\frac{x + 4}{3} \]This gives the inverse function: \[ f^{-1}(x) = -\frac{x + 4}{3} \]

Step 4: Verify \((f \circ f^{-1})(x) = x\)

Substitute \(f^{-1}(x)\) into \(f(x)\):\[ f(f^{-1}(x)) = -3\left(-\frac{x + 4}{3}\right) - 4 \]Simplify this:\[ = (x + 4) - 4 = x \]Thus, \((f \circ f^{-1})(x) = x\).

Step 5: Verify \((f^{-1} \circ f)(x) = x\)

Substitute \(f(x)\) into \(f^{-1}(x)\):\[ f^{-1}(f(x)) = -\frac{(-3x - 4) + 4}{3} \]Simplify:\[ = -\frac{-3x}{3} = x \]Thus, \((f^{-1} \circ f)(x) = x\).

Key Concepts

Function Composition

Function composition is an operation that takes two functions and combines them to form a new function. It is like plugging one function into another. When you see \((f \circ g)(x)\), it means you take the output of function \(g\) and use it as the input for function \(f\). This notation can be a bit abstract at first, but with a bit of practice, you'll find that it's a powerful tool.

• Order Matters: Remember, the order in which you compose functions is crucial. \(f \circ g\) is not the same as \(g \circ f\).
• Combining Functions: Function composition allows us to see how changes or operations in one part affect the overall outcome.

For the exercise given, to find out that \(f(f^{-1}(x))=x\), you're effectively running two operations that cancel each other out, bringing us back to our starting point, \(x\). That's why it's commonly used in the verification of inverse functions.

Algebraic Manipulation

Algebraic manipulation involves rewriting mathematical expressions or equations for various purposes. This is a vital skill when finding the inverse of a function. It typically requires the following steps:

• Start with replacing \(f(x)\) with \(y\): This often represents our function setup, such as \[y = -3x-4\]
• Isolating Variables: Reorder the equation to isolate the variable of interest, in this case, \(x\), by using algebraic operations such as addition, subtraction, multiplication, or division.
• Swapping Roles: Once we have \(x\) expressed in terms of \(y\), we swap these variables to express the inverse, giving us \(f^{-1}(x)\).

By practicing these steps, you build an intuition for how functions and their inverses are connected. This manipulation is essential for understanding relationships within mathematical contexts.

Verification of Inverse Functions

Verification of inverse functions is crucial for confirming that two functions indeed undo each other's actions. To check this, you need to perform two compositions:1. \(f \circ f^{-1} = x\)2. \(f^{-1} \circ f = x\)Each composition should simplify to \(x\), illustrating they revert each other's effects.

• Substitution: In the problem, you substitute \(f^{-1}(x)\) into \(f(x)\), simplify, and verify if it returns \(x\).
• Double-Check: Similarly, substitute \(f(x)\) into \(f^{-1}(x)\) and check if you get \(x\).

Verification confirms that your derived inverse function is correct. It reassures that you've performed algebra manipulations and function compositions accurately. With this thorough checking process, you gain confidence in your understanding and the validity of your solutions.