A one-to-one function is given. Write an equation for the inverse of the function.
[tex]$f(x)=\frac{7-x}{3}$[/tex]
Answer (2)
Replace f ( x ) with y : y = 3 7 − x .
Swap x and y : x = 3 7 − y .
Solve for y : y = 7 − 3 x .
Replace y with f − 1 ( x ) : f − 1 ( x ) = 7 − 3 x , so f − 1 ( x ) = 7 − 3 x .
Explanation
Understanding the Problem We are given a one-to-one function f ( x ) = f r a c 7 − x 3 and we want to find its inverse function, denoted as f − 1 ( x ) . The inverse function essentially 'undoes' what the original function does.
Finding the Inverse Function To find the inverse function, we'll follow these steps:
Replace f ( x ) with y : y = f r a c 7 − x 3 .
Swap x and y : x = f r a c 7 − y 3 .
Solve for y in terms of x .
Isolating y Let's solve for y :
Starting with x = f r a c 7 − y 3 , we multiply both sides of the equation by 3 to get rid of the fraction:
3 x = 7 − y
Next, we want to isolate y . We can add y to both sides and subtract 3 x from both sides:
y = 7 − 3 x
Writing the Inverse Function Now that we have solved for y , we replace y with f − 1 ( x ) to denote the inverse function:
f − 1 ( x ) = 7 − 3 x
Final Answer Therefore, the inverse function of f ( x ) = f r a c 7 − x 3 is f − 1 ( x ) = 7 − 3 x .
Examples
Imagine you are converting temperatures from Celsius to Fahrenheit using a function. The inverse function would then convert temperatures from Fahrenheit back to Celsius. In general, inverse functions are useful in any situation where you need to 'undo' a process or reverse a transformation. For example, if you encode a message using a function, the inverse function would decode the message back to its original form. This concept is widely used in cryptography, computer science, and engineering.
The inverse function of f ( x ) = 3 7 − x is found by swapping and solving for y . The final result is f − 1 ( x ) = 7 − 3 x .
;
