If $g(x)$ is the inverse of $f(x)$ and $f(x)=4x+12$, what is $g(x)$?

A. $g(x)=12x+4$
B. $g(x)=\frac{1}{4}x-12$
C. $g(x)=x-3$
D. $g(x)=\frac{1}{4}x-3

The inverse function of f ( x ) = 4 x + 12 is g ( x ) = 4 1 ​ x − 3 . Therefore, the correct answer is option D. This function effectively reverses the operation of the original function, allowing you to find the input from the output of f ( x ) .
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Answered by Anonymous | 2025-07-04

Replace f ( x ) with y : y = 4 x + 12 .
Swap x and y : x = 4 y + 12 .
Solve for y : y = f r a c x − 12 4 = f r a c 1 4 x − 3 .
Replace y with g ( x ) : g ( x ) = f r a c 1 4 x − 3 . The inverse function is f r a c 1 4 x − 3 ​ .

Explanation

Understanding the Problem We are given that g ( x ) is the inverse of f ( x ) , and f ( x ) = 4 x + 12 . Our goal is to find the expression for g ( x ) . The inverse of a function is found by swapping the roles of x and y and then solving for y .

Replacing f(x) with y First, let's write the function f ( x ) as an equation with y : y = 4 x + 12

Swapping x and y Next, we swap x and y :
x = 4 y + 12

Isolating the term with y Now, we solve for y . Subtract 12 from both sides: x − 12 = 4 y

Solving for y Divide both sides by 4: y = f r a c x − 12 4 We can simplify this expression: y = f r a c x 4 − f r a c 12 4 y = f r a c 1 4 x − 3

Finding the Inverse Function Finally, we replace y with g ( x ) to denote the inverse function: g ( x ) = f r a c 1 4 x − 3 So, the inverse function is g ( x ) = 4 1 ​ x − 3 .


Examples
Imagine you're converting temperatures between Celsius and Fahrenheit. If f ( x ) converts Celsius to Fahrenheit, then g ( x ) would convert Fahrenheit back to Celsius. Similarly, if f ( x ) calculates the total cost of buying x items at a certain price plus a fixed shipping fee, then g ( x ) would calculate how many items you can buy with a given amount of money. Understanding inverse functions helps reverse processes and solve for the original input.

Answered by GinnyAnswer | 2025-07-04