If [tex]f(x)=\frac{5 x}{3}+5[/tex], which of the following is the inverse of [tex]f(x)[/tex] ?
A. [tex]f^{-1}(x)=\frac{5(x+5)}{3}[/tex]
B. [tex]f^{-1}(x)=\frac{3(x-5)}{5}[/tex]
C. [tex]f^{-1}(x)=\frac{5(x-5)}{3}[/tex]
D. [tex]f^{-1}(x)=\frac{3(x+5)}{5}[/tex]
Answer (1)
Replace f ( x ) with y : y = 3 5 x + 5 .
Swap x and y : x = 3 5 y + 5 .
Solve for y : y = 5 3 ( x − 5 ) .
The inverse function is f − 1 ( x ) = 5 3 ( x − 5 ) , which corresponds to option B. f − 1 ( x ) = 5 3 ( x − 5 )
Explanation
Understanding the Problem We are given the function f ( x ) = 3 5 x + 5 and we want to find its inverse, denoted as f − 1 ( x ) . The inverse function essentially reverses the operation of the original function.
Replace f(x) with y To find the inverse function, we first replace f ( x ) with y , so we have: y = 3 5 x + 5
Swap x and y Next, we swap x and y :
x = 3 5 y + 5
Isolate the term with y Now, we solve for y in terms of x . First, subtract 5 from both sides: x − 5 = 3 5 y
Multiply by 3 Multiply both sides by 3: 3 ( x − 5 ) = 5 y
Divide by 5 Finally, divide both sides by 5: y = 5 3 ( x − 5 )
Write the inverse function So, the inverse function is: f − 1 ( x ) = 5 3 ( x − 5 )
Final Answer Comparing this with the given options, we see that it matches option B.
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 back to Celsius. Understanding inverse functions helps in reversing processes or calculations, which is useful in many scientific and engineering applications.
