Which function is the inverse of [tex]f(x)=2 x+3[/tex] ?
A. [tex]f^{-1}(x)=-\frac{1}{2} x-\frac{3}{2}[/tex]
B. [tex]f^{-1}(x)=\frac{1}{2} x-\frac{3}{2}[/tex]
C. [tex]f^{-1}(x)=-2 x+3[/tex]
D. [tex]f^{-1}(x)=2 x+3[/tex]
Answer (1)
Replace f ( x ) with y : y = 2 x + 3 .
Swap x and y : x = 2 y + 3 .
Solve for y : y = 2 x − 3 = 2 1 x − 2 3 .
The inverse function is f − 1 ( x ) = 2 1 x − 2 3 .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 x + 3 and we want to find its inverse, f − 1 ( x ) . The inverse function is found by swapping x and y and then solving for y .
Replace f(x) with y First, replace f ( x ) with y : y = 2 x + 3
Swap x and y Next, swap x and y : x = 2 y + 3
Isolate the term with y Now, solve for y in terms of x . Subtract 3 from both sides: x − 3 = 2 y
Solve for y Divide both sides by 2: y = 2 x − 3
Rewrite the equation Rewrite as: y = 2 1 x − 2 3
Write the inverse function Finally, replace y with f − 1 ( x ) : f − 1 ( x ) = 2 1 x − 2 3
Final Answer Therefore, the inverse function is f − 1 ( x ) = 2 1 x − 2 3 .
Examples
Imagine you're converting temperatures from Celsius to Fahrenheit using the formula F = 5 9 C + 32 . Finding the inverse function allows you to convert from Fahrenheit back to Celsius. This is useful in many real-world situations, such as when traveling to different countries or when working with scientific data that uses different temperature scales. Understanding inverse functions helps you reverse processes and solve for the original input.
