What is the inverse of the function [tex]f(x)=4 x+8[/tex]?
A. [tex]h(x)=\frac{1}{4} x-2[/tex]
B. [tex]h(x)=\frac{1}{4} x+2[/tex]
C. [tex]h(x)=\frac{1}{2} x-2[/tex]
D. [tex]h(x)=\frac{1}{2} x+2[/tex]
Answer (2)
Replace f ( x ) with y : y = 4 x + 8 .
Swap x and y : x = 4 y + 8 .
Solve for y : y = 4 x − 8 .
Simplify: h ( x ) = 4 1 x − 2 , so the inverse function is h ( x ) = 4 1 x − 2 .
Explanation
Understanding the Problem We are given the function f ( x ) = 4 x + 8 and we want to find its inverse, which we'll call h ( x ) . The inverse function essentially 'undoes' what the original function does.
Replacing f(x) with y To find the inverse, we start by writing the function as y = 4 x + 8 .
Swapping x and y Next, we swap x and y to get x = 4 y + 8 . This is the key step in finding the inverse.
Isolating the term with y Now, we solve for y in terms of x . First, subtract 8 from both sides of the equation: x − 8 = 4 y
Solving for y Then, divide both sides by 4: 4 x − 8 = y
Rewriting the equation We can rewrite this as: y = 4 x − 4 8
Simplifying the fraction Simplifying the fraction, we get: y = 4 1 x − 2
The Inverse Function So, the inverse function is h ( x ) = 4 1 x − 2 .
Final Answer Therefore, the inverse of the function f ( x ) = 4 x + 8 is h ( x ) = 4 1 x − 2 .
Examples
Imagine you're converting temperatures from Celsius to Fahrenheit using the formula F = 5 9 C + 32 . Finding the inverse function would allow you to convert from Fahrenheit back to Celsius. In general, inverse functions are useful in any situation where you need to reverse a process or calculation. For example, if you have a code that encrypts a message, the inverse function would be the decryption code.
The inverse of the function f ( x ) = 4 x + 8 is h ( x ) = 4 1 x − 2 . This means that option A is the correct choice. We arrived at this conclusion by swapping the variables and solving for y .
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