Which function is the inverse of [tex]f(x)=-5 x-4[/tex] ?

A. [tex]f^{-1}(x)=-\frac{1}{5} x-\frac{4}{5}[/tex]
B. [tex]f^{-1}(x)=-\frac{1}{5} x+\frac{4}{5}[/tex]
C. [tex]f^{-1}(x)=-4 x+5[/tex]
D. [tex]f^{-1}(x)=4 x+4[/tex]

Replace f ( x ) with y : y = − 5 x − 4 .
Swap x and y : x = − 5 y − 4 .
Solve for y : y = − 5 x + 4 ​ .
Rewrite and express the inverse function: f − 1 ( x ) = − 5 1 ​ x − 5 4 ​ .

Explanation

Understanding the Inverse Function To find the inverse of the function f ( x ) = − 5 x − 4 , we need to switch the roles of x and y and then solve for y . This will give us the inverse function, denoted as f − 1 ( x ) .

Replacing f(x) with y First, let's replace f ( x ) with y : y = − 5 x − 4

Swapping x and y Next, we swap x and y : x = − 5 y − 4

Isolating the term with y Now, we solve for y . Add 4 to both sides of the equation: x + 4 = − 5 y

Solving for y Divide both sides by -5: y = − 5 x + 4 ​

Rewriting the equation We can rewrite this as: y = − 5 1 ​ x − 5 4 ​

The Inverse Function Finally, we replace y with f − 1 ( x ) : f − 1 ( x ) = − 5 1 ​ x − 5 4 ​ So, the inverse function is f − 1 ( x ) = − 5 1 ​ x − 5 4 ​ .


Examples
Imagine you're converting temperatures from Celsius to Fahrenheit using a function. The inverse function would then convert Fahrenheit back to Celsius. Similarly, if you have a function that encodes a message, the inverse function would decode it. In general, inverse functions are useful for reversing processes or undoing operations, which is a common task in many areas of math and science.

Answered by GinnyAnswer | 2025-07-03

The inverse function of f ( x ) = − 5 x − 4 is f − 1 ( x ) = − 5 1 ​ x − 5 4 ​ . Therefore, the correct option is A. This process involves replacing f(x) with y, swapping x and y, and solving for y.
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Answered by Anonymous | 2025-07-04