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 (2)
Replace f ( x ) with y : y = 2 x + 3 .
Swap x and y : x = 2 y + 3 .
Solve for y : y = 2 x − 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 need 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
Write the inverse function Rewrite the inverse function: f − 1 ( x ) = 2 x − 3 = 2 1 x − 2 3
Final Answer Comparing our result with the given options, we see that the correct answer is f − 1 ( x ) = 2 1 x − 2 3 .
Examples
In real life, inverse functions can be used to convert between different units of measurement. For example, if f ( x ) converts Celsius to Fahrenheit, then f − 1 ( x ) converts Fahrenheit to Celsius. Understanding inverse functions helps in reversing processes and conversions, which is useful in many scientific and engineering applications.
The inverse of the function f ( x ) = 2 x + 3 is found by solving for y after swapping it with x . The resulting inverse function is f − 1 ( x ) = 2 1 x − 2 3 , which corresponds to option B. Thus, the inverse is B: f − 1 ( x ) = 2 1 x − 2 3 .
;
