The relationship between a salesperson's bonus, in thousands of dollars, [tex]$f(x)$[/tex], and the number of cars sold, [tex]$x$[/tex], is given by [tex]$f(x)=10 x+24$[/tex]. Which function represents the number of cars sold in terms of the bonus received?
A. [tex]$f^{-1}(x)=\frac{x-24}{10}$[/tex]
B. [tex]$f^{-1}(x)=10(x-24)$[/tex]
C. [tex]$f^{-1}(x)=\frac{x}{10}-24$[/tex]
D. [tex]$f^{-1}(x)=10 x-24$[/tex]
Answer (1)
To find the inverse function f − 1 ( x ) :
Replace f ( x ) with y : y = 10 x + 24 .
Swap x and y : x = 10 y + 24 .
Solve for y : y = 10 x − 24 .
Replace y with f − 1 ( x ) : f − 1 ( x ) = 10 x − 24 .
The function representing the number of cars sold in terms of the bonus received is f − 1 ( x ) = 10 x − 24 .
Explanation
Understanding the Problem We are given the function f ( x ) = 10 x + 24 , which represents the relationship between a salesperson's bonus, f ( x ) , in thousands of dollars, and the number of cars sold, x . We need to find the inverse function, f − 1 ( x ) , which represents the number of cars sold in terms of the bonus received.
Finding the Inverse Function To find the inverse function, we can follow these steps:
Replace f ( x ) with y : y = 10 x + 24 .
Swap x and y : x = 10 y + 24 .
Solve for y in terms of x :
Subtract 24 from both sides: x − 24 = 10 y .
Divide both sides by 10: y = 10 x − 24 .
Replace y with f − 1 ( x ) : f − 1 ( x ) = 10 x − 24 .
Final Answer Therefore, the function that represents the number of cars sold in terms of the bonus received is f − 1 ( x ) = 10 x − 24 .
Examples
Understanding inverse functions is crucial in many real-world scenarios. For instance, if you know how much time it takes to bake a cake based on the oven temperature, the inverse function would tell you what temperature to set the oven to achieve a specific baking time. Similarly, in economics, if you know how the price of a product affects its demand, the inverse function would tell you what price to set to achieve a specific demand level. These concepts are fundamental in optimizing processes and making informed decisions.
