Select the correct answer.
If $f(x)=5 x-3$, which expression represents its inverse function?
A. $\quad \frac{(x+3)}{(x+2)}$
B. $\frac{(x+3)}{(5)}$
C. $\frac{(5)}{(x+3)}$
D. $\frac{(x-3)}{(2)}$
Answer (1)
Replace f ( x ) with y : y = 5 x − 3 .
Swap x and y : x = 5 y − 3 .
Solve for y : y = 5 x + 3 .
The inverse function is f − 1 ( x ) = 5 x + 3 , so the answer is ( 5 ) ( x + 3 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = 5 x − 3 and we want to find its inverse function, denoted as f − 1 ( x ) . The inverse function essentially reverses the operation of the original function.
Finding the Inverse To find the inverse function, we can follow these steps:
Replace f ( x ) with y : y = 5 x − 3 .
Swap x and y : x = 5 y − 3 .
Solve for y in terms of x .
Solving for y Let's solve for y :
x = 5 y − 3
Add 3 to both sides:
x + 3 = 5 y
Divide both sides by 5:
y = 5 x + 3
The Inverse Function Now, replace y with f − 1 ( x ) :
f − 1 ( x ) = 5 x + 3
Final Answer Comparing our result with the given options, we see that option B matches our inverse function.
Therefore, the correct answer is B.
Examples
Imagine you're baking a cake and need to convert Celsius to Fahrenheit using the formula F = 5 9 C + 32 . If you know the Fahrenheit temperature and need to find the Celsius temperature, you'd use the inverse function C = 9 5 ( F − 32 ) . Similarly, inverse functions are used in cryptography to decode messages, in economics to reverse supply and demand equations, and in computer graphics to transform coordinates back to their original positions.
