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)
Write the function as an equation: y = f r a c 1 4 x − 12 .
Swap x and y : x = f r a c 1 4 y − 12 .
Solve for y : y = 4 x + 48 .
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'll call h ( x ) . The inverse function essentially 'undoes' what the original function does.
Writing the Equation To find the inverse, we start by writing the function as an equation: y = 4 1 x − 12
Swapping x and y Next, we swap x and y to get: x = 4 1 y − 12
Isolating y Now, we solve for y . First, add 12 to both sides of the equation: x + 12 = 4 1 y
Solving for y Then, multiply both sides by 4 to isolate y : 4 ( x + 12 ) = y 4 x + 48 = y
The Inverse Function So, the inverse function is: h ( x ) = 4 x + 48
Final Answer Therefore, the inverse of the function f ( x ) = 4 1 x − 12 is h ( x ) = 4 x + 48 .
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 calculates the cost of an item after a discount, the inverse function would calculate the original price before the discount. Understanding inverse functions helps in reversing processes and calculations in various real-life scenarios.
The inverse of the function f ( x ) = 4 1 x − 12 is found to be h ( x ) = 4 x + 48 , which matches option D. To find the inverse, we swapped x and y and solved for y step by step. This confirmed that the correct option is D: h ( x ) = 4 x + 48 .
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