Problem 24 - Intermediate Algebra

Question

Find the inverse of the given function by using the "undoing process," and then verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). (Objective 4) $$f(x)=-4 x-3$$

Step-by-Step Solution

Answer

The inverse of \(f(x) = -4x - 3\) is \(f^{-1}(x) = -\frac{x+3}{4}\).

Step 1: Understanding the Function

We are given the function \(f(x) = -4x - 3\). The goal is to find the inverse, meaning we want a function that "undoes" the effect of \(f(x)\).

Step 2: Substitute and Solve for x

To find the inverse, we start with \(y = -4x - 3\). Switch \(x\) and \(y\) and solve for \(y\):1. Swap \(x\) and \(y\): \(x = -4y - 3\).2. Solve for \(y\): - Add 3 to both sides: \(x + 3 = -4y\). - Divide by -4: \(y = -\frac{x+3}{4}\). This gives us the inverse function \(f^{-1}(x) = -\frac{x+3}{4}\).

Step 3: Verify Inverse Composition 1

To verify \((f \circ f^{-1})(x) = x\), substitute \(f^{-1}(x) = -\frac{x+3}{4}\) into \(f(x)\):\( f(f^{-1}(x)) = f\left(-\frac{x+3}{4}\right) = -4\left(-\frac{x+3}{4}\right) - 3\).Simplify the expression:\( = (x+3) - 3 = x\).This confirms that \((f \circ f^{-1})(x) = x\).

Step 4: Verify Inverse Composition 2

To verify \((f^{-1} \circ f)(x) = x\), substitute \(f(x) = -4x - 3\) into \(f^{-1}(x)\):\( f^{-1}(f(x)) = f^{-1}(-4x - 3) = -\frac{(-4x - 3) + 3}{4}\).Simplify the expression:\( = -\frac{-4x}{4} = x\).This confirms that \((f^{-1} \circ f)(x) = x\).

Key Concepts

Function Composition

Function composition is an essential concept when working with inverse functions. It involves creating a new function by applying one function to the result of another. In symbols, function composition of two functions, say \( f \) and \( g \), is represented as \( (f \circ g)(x) \), which means \( f(g(x)) \). This is a critical factor in verifying if two functions are inverses of each other.

• When two functions \( f \) and \( g \) are inverses, the composition \( (f \circ g)(x) \) should equal \( x \), and vice versa, \( (g \circ f)(x) \) should also equal \( x \).
• This shows that applying \( g \) after \( f \), or \( f \) after \( g \), returns us back to our starting value \( x \).

Understanding this dual condition is crucial when verifying inverse relationships in algebra. Function composition tells us that no matter which order we apply these inverse functions, the original input is retrieved.

Algebraic Manipulation

Algebraic manipulation refers to the techniques used to rearrange equations or expressions in order to solve for a desired variable. It allows us to find the inverse of a function by switching dependent and independent variables, then isolating the new dependent variable. Let’s break it down:

• Start by replacing \( f(x) \) with \( y \) in the original function: \( y = -4x - 3 \).
• Swap \( x \) and \( y \). This is the cornerstone of finding an inverse, turning \( x \) into a function of \( y \), resulting in \( x = -4y - 3 \).
• Rearrange the equation to isolate \( y \): Move terms and divide by coefficients as needed, which provides us \( y = -\frac{x+3}{4} \).

This process yields the inverse function \( f^{-1}(x) = -\frac{x+3}{4} \). Algebraic manipulation makes finding the inverse possible by carefully adjusting each part of the function until the solution is clear.

Verification of Inverse

Verification of inverse functions ensures that the inverse function derived is correct. This involves composition of the function and its inverse, with two main verifications:

• First, check \( (f \circ f^{-1})(x) = x \). Begin by substituting the inverse function into the original function. Simplifying the equation should bring you back to the variable \( x \).
• Second, check \( (f^{-1} \circ f)(x) = x \). This is done by applying the original function within the inverse function. Once again, the outcome should simplify back to \( x \).

If both conditions are satisfied, then the functions are indeed inverses. This dual testing of composition acts as a thorough check, ensuring no errors were made in algebraic manipulation. Successfully verifying these compositions shows complete certainty in the function pairs being true mathematical inverses.