Problem 33 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=4 x+7\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{x - 7}{4}\).

Step 1: Understand the Definition

An inverse function \(f^{-1}(x)\) is a function that reverses the effect of \(f(x)\). This means that \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(x)) = x\). To find \(f^{-1}(x)\), we need to solve the equation \(y = f(x)\) for \(x\).

Step 2: Replace Function Notation

Replace \(f(x)\) with \(y\) in the equation, so we have the equation \(y = 4x + 7\).

Step 3: Solve for \(x\)

Rearrange the equation \(y = 4x + 7\) to solve for \(x\). This involves isolating \(x\) on one side of the equation. Begin by subtracting 7 from both sides: \(y - 7 = 4x\).

Step 4: Solve for \(x\) Continued

Complete the isolation of \(x\) by dividing both sides of the equation by 4: \(x = \frac{y - 7}{4}\).

Step 5: Define the Inverse Function

Replace \(y\) back with \(x\) to find the inverse function. The expression for the inverse function is \(f^{-1}(x) = \frac{x - 7}{4}\).

Step 6: Verify the Inverse Function

To ensure correctness, verify by checking \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). 1. Calculate \(f(f^{-1}(x))\): - \(f\left(\frac{x - 7}{4}\right) = 4\left(\frac{x - 7}{4}\right) + 7 = x - 7 + 7 = x\).2. Calculate \(f^{-1}(f(x))\): - \(f^{-1}(4x + 7) = \frac{4x + 7 - 7}{4} = \frac{4x}{4} = x\).Both compositions return \(x\), confirming the inverse function.

Key Concepts

Function Composition

Function composition is a vital concept when working with inverse functions. It's the process of combining two functions, where the output of one function becomes the input of the next. For inverse functions specifically, this helps verify that we have correctly found the inverse.

• When two functions, say \( f(x) \) and \( g(x) \), are composed, we write \( (f \circ g)(x) = f(g(x)) \). This means you first apply \( g \) to \( x \), and then apply \( f \) to the result.
• For inverse functions, if \( f^{-1}(x) \) is truly the inverse, then \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). These are essential checks.

This exercise involves verifying the inverse \( f^{-1}(x) = \frac{x - 7}{4} \) by checking these compositions. When you compute \( f(f^{-1}(x)) \) and get back \( x \), and similarly with \( f^{-1}(f(x)) \), it confirms the correctness of the inverse function.

Algebraic Manipulation

Algebraic manipulation is at the core of solving equations to find inverse functions. The goal is to express one variable in terms of another by rearranging the given formula.

• Consider the original equation \( y = 4x + 7 \). To find the inverse, we start by treating \( y \) as \( f(x) \) and aim to rewrite the equation to solve for \( x \).
• Subtract 7 from both sides to get \( y - 7 = 4x \). This step involves isolating the terms that include \( x \).
• Divide by 4 to solve for \( x \): \( x = \frac{y - 7}{4} \). Each algebraic step should preserve the equality and aim to make \( x \) appear by itself.

Mastering algebraic manipulation is key to finding and verifying inverse functions. In this problem, these actions transform the expression such that it offers a formula computing the input of the original function.

Linear Functions

Linear functions like \( f(x) = 4x + 7 \) have a simple form that makes finding inverse functions relatively straightforward. These functions follow the rule \( f(x) = mx + b \), where \( m \) and \( b \) are constants.

• In a linear function, \( m \) is the slope and \( b \) is the y-intercept. The slope tells you how steep the line is, while the y-intercept tells you where the line crosses the y-axis.
• Because linear functions have consistent rates of change, their inverses maintain a basic form. The inverse of a linear function will also be linear, as shown with \( f^{-1}(x) = \frac{x - 7}{4} \).

Understanding the properties of linear functions helps simplify the process when solving for inverses. Since each algebraic step leaves the linearity intact, even complex calculations remain manageable. Focusing on the slope and intercept gives insight into the function's behavior both before and after taking its inverse.