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Chapter 10: Problem 48

In the following exercises, find the inverse of each function. $$ f(x)=-5 x-4 $$

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = -\frac{x + 4}{5} \).

Step by step solution

01

- Rewrite the function

First, write the given function as an equation: \[ y = -5x - 4 \]
02

- Swap variables

To find the inverse, swap the roles of x and y: \[ x = -5y - 4 \]
03

- Solve for y

Now, solve for y in terms of x. First, add 4 to both sides: \[ x + 4 = -5y \] Then, divide both sides by -5: \[ y = -\frac{x+4}{5} \]
04

- Write the inverse function

Finally, replace y with \( f^{-1}(x) \): \[ f^{-1}(x) = -\frac{x + 4}{5} \]

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Key Concepts

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

Function Inversion
In mathematics, finding the inverse of a function involves reversing the process of the original function. Essentially, if you have a function that maps an input to an output, the inverse function takes that output and maps it back to the original input. In simpler terms, if the function is a machine that processes inputs to outputs, the inverse function is the machine that can take those outputs and determine what the original inputs were.
Algebraic Manipulation
Algebraic manipulation is a crucial step when finding the inverse of a function. This involves rearranging the original equation to isolate the variable of interest. Here are a few tricks for effective algebraic manipulation:
  • Use addition or subtraction to move terms from one side of the equation to the other.
  • Apply multiplication or division to isolate the variable.
  • Use parentheses to group terms and simplify the process.
For example, when we had the equation \( y = -5x - 4 \), we performed a series of algebraic manipulations to solve for \( y \). These steps involve rearranging terms and simplifying equations.
Solving Equations
Solving equations is essential not just for inverse functions but for many mathematical tasks. The goal is to find the value of the variable that makes the equation true. There are several methods to solve equations, such as:
  • Addition/Subtraction: Adding or subtracting the same number from both sides of the equation.
  • Multiplication/Division: Multiplying or dividing both sides by the same number to isolate the variable.
  • Combining like terms: Simplifying expressions to make the equation easier to solve.
For instance, in step 3 of the problem, where we had \( x + 4 = -5y \), we performed both addition and division to solve for \( y \).
Switching Variables
Switching variables is a key step when finding an inverse function. This technique involves swapping the dependent and independent variables to reflect the reverse process of the function. Here's how it works:
  • Begin by writing the function in terms of \( y \): \( y = -5x - 4 \).
  • Swap the variables \( x \) and \( y \): \( x = -5y - 4 \).
Once the variables are switched, you can perform algebraic manipulations to isolate the new dependent variable, as you solve the modified equation.

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