If [tex]f(x)=\frac{2 x-3}{5}[/tex], which of the following is the inverse of [tex]f(x)[/tex]?
A. [tex]f^{-1}(x)=\frac{3 x+2}{5}[/tex]
B. [tex]f^{-1}(x)=\frac{5 x+3}{2}[/tex]
C. [tex]f^{-1}(x)=\frac{3 x+5}{2}[/tex]
D. [tex]f^{-1}(x)=\frac{2 x+3}{5}[/tex]
Answer (1)
Replace f ( x ) with y in the original equation.
Swap x and y to prepare for finding the inverse.
Solve the equation for y in terms of x .
The inverse function is f − 1 ( x ) = 2 5 x + 3 .
Explanation
Understanding the Problem We are given the function f ( x ) = 5 2 x − 3 and we want to find its inverse, denoted as f − 1 ( x ) . The inverse function essentially 'undoes' what the original function does.
Finding the Inverse To find the inverse, we'll follow these steps:
Replace f ( x ) with y : y = 5 2 x − 3 .
Swap x and y : x = 5 2 y − 3 .
Solve for y in terms of x .
Solving for y Let's solve for y :
Multiply both sides of the equation x = 5 2 y − 3 by 5 to get rid of the fraction: 5 x = 2 y − 3 Add 3 to both sides: 5 x + 3 = 2 y Divide both sides by 2: y = 2 5 x + 3 So, the inverse function is f − 1 ( x ) = 2 5 x + 3 .
Comparing with Options Now, we compare our result with the given options:
A. f − 1 ( x ) = 5 3 x + 2 B. f − 1 ( x ) = 2 5 x + 3 C. f − 1 ( x ) = 2 3 x + 5 D. f − 1 ( x ) = 5 2 x + 3
Our calculated inverse function matches option B.
Final Answer Therefore, the inverse of the function f ( x ) = 5 2 x − 3 is f − 1 ( x ) = 2 5 x + 3 .
Examples
Imagine you have a machine that converts kilograms to pounds using the function f ( x ) = 5 2 x − 3 (where this function is just an example and not a real conversion). Finding the inverse function allows you to convert pounds back to kilograms. This is useful in many real-world scenarios where you need to reverse a process or calculation. For instance, if a recipe is given in grams but you have a scale that measures in ounces, you'd use an inverse function to convert the measurements.
