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Chapter 10: Problem 45

In the following exercises, find the inverse of each function. $$ f(x)=6 x-7 $$

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = \frac{x + 7}{6} \).

Step by step solution

01

Replace f(x) with y

First, write the given function as an equation where \( f(x) = 6x - 7 \) becomes \( y = 6x - 7 \).
02

Swap x and y

To find the inverse, swap the variables x and y in the equation $$ y = 6x - 7 \rightarrow x = 6y - 7 $$.
03

Solve for y

Isolate y to solve the equation for y: $$ x = 6y - 7 $$ $$ x + 7 = 6y $$ $$ y = \frac{x + 7}{6} $$.
04

Rewrite as the inverse function

Express the function as the inverse: $$ f^{-1}(x) = \frac{x + 7}{6} $$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Functions
Algebraic functions are expressions that relate inputs to outputs using basic arithmetic. In our example, the function given is: \( f(x) = 6x - 7 \).

A few key points to understand about algebraic functions:
  • Linear Functions: These are functions where the highest power of the variable (x) is one, just like in our function above.
  • Operations: Linear functions can involve basic operations such as addition, subtraction, multiplication, and division.
  • Behavior: When plotted on a graph, a linear function shows a straight line, indicating a constant rate of change.
Understanding these basics helps in comprehending how functions behave and how to manipulate them.
Solving Equations
Solving equations is a central part of finding the inverse function. When we need to solve for a variable, we perform operations to isolate that variable. Let's look at the steps involved in solving our specific equation to find its inverse.

  • Initial Setup: Start with the equation derived from the function: \( y = 6x - 7 \).
  • Swap Variables: To find the inverse, swap x and y: \( x = 6y - 7 \).
  • Isolate the Variable: Solve for y by performing algebraic operations:
    \( x = 6y - 7 \)
    \( x + 7 = 6y \)
    \( y = \frac{x + 7}{6} \).

By isolating y, we determine the inverse function, illustrating the importance of each algebraic step in achieving the solution.
Function Notation
Function notation is a way to represent functions clearly and concisely. The given function is written as \( f(x) \), where f represents the function, and x is the input variable.

When finding the inverse, the notation changes to \( f^{-1}(x) \). This indicates that the function reverses the roles of inputs and outputs. Here's how you can understand it better:

  • Original Function: For \( f(x) = 6x - 7 \), input x gives a specific output.
  • Inverse Function: For \( f^{-1}(x) = \frac{x + 7}{6} \), inputting the output from the original function retrieves the original input.
  • Notation: The notation helps distinguish between the function and its inverse, making it easier to understand and communicate the relationships between variables.

Grasping function notation is essential for solving equations and finding inverses, as it provides a clear framework for expressing these relationships.

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Most popular questions from this chapter

In the following exercises, solve for \(x\). \(\log (x-1)-\log (x+3)=\log \frac{1}{x}\)

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