What is the inverse of the function [tex]f(x)=\frac{1}{4} x-12[/tex]?
A. [tex]h(x)=48 x-4[/tex]
B. [tex]h(x)=48 x+4[/tex]
C. [tex]h(x)=4 x-48[/tex]
D. [tex]h(x)=4 x+48[/tex]
Answer (2)
Replace f ( x ) with y : y = 4 1 x − 12 .
Swap x and y : x = 4 1 y − 12 .
Solve for y : y = 4 x + 48 .
Replace y with h ( x ) : h ( x ) = 4 x + 48 , so the inverse function is h ( x ) = 4 x + 48 .
Explanation
Understanding the Problem We are given the function f ( x ) = f r a c 1 4 x − 12 and we want to find its inverse, which we will call h ( x ) . The inverse function is found by swapping x and y and solving for y .
Replace f(x) with y First, replace f ( x ) with y : y = 4 1 x − 12
Swap x and y Next, swap x and y : x = 4 1 y − 12
Add 12 to both sides Now, solve for y in terms of x . Add 12 to both sides of the equation: x + 12 = 4 1 y
Multiply both sides by 4 Multiply both sides by 4 to isolate y : 4 ( x + 12 ) = y
Simplify Distribute the 4: 4 x + 48 = y
Replace y with h(x) Finally, replace y with h ( x ) to denote the inverse function: h ( x ) = 4 x + 48
Final Answer Therefore, the inverse of the function f ( x ) = f r a c 1 4 x − 12 is h ( x ) = 4 x + 48 .
Examples
Understanding inverse functions is crucial in many real-world applications. For example, if f ( x ) represents the cost of producing x items, then the inverse function h ( x ) would represent the number of items that can be produced for a cost of x . In this case, if the cost of producing x items is given by f ( x ) = f r a c 1 4 x − 12 , then the number of items that can be produced for a cost of x is given by h ( x ) = 4 x + 48 . This concept is widely used in economics, engineering, and computer science to reverse processes and solve for unknown variables.
The inverse of the function f ( x ) = 4 1 x − 12 is h ( x ) = 4 x + 48 . Therefore, the correct option is D. This can be derived by swapping x and y and solving for y .
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