Select the correct answer.
Which of the following is the inverse of [tex]$f(x)=\frac{3-x}{5}$[/tex]?
A. [tex]$f^{-1}(x)=5 x-3$[/tex]
B. [tex]$f^{-1}(x)=\frac{5+x}{3}$[/tex]
C. [tex]$f^{-1}(x)=3-\frac{x}{5}$[/tex]
D. [tex]$f^{-1}(x)=3-5 x$[/tex]
Answer (2)
Replace f ( x ) with y : y = 5 3 − x .
Swap x and y : x = 5 3 − y .
Solve for y : 5 x = 3 − y , which gives y = 3 − 5 x .
The inverse function is f − 1 ( x ) = 3 − 5 x .
Explanation
Understanding the Problem We are given the function f ( x ) = 5 3 − x and we want to find its inverse, f − 1 ( x ) . The inverse function is found by swapping x and y and solving for y .
Replacing f(x) with y Let y = f ( x ) , so we have y = 5 3 − x .
Swapping x and y Now, we swap x and y to get x = 5 3 − y .
Solving for y - Step 1 Next, we solve for y . Multiply both sides of the equation by 5: 5 x = 3 − y
Solving for y - Step 2 Add y to both sides and subtract 5 x from both sides: y = 3 − 5 x
Finding the Inverse Function Therefore, the inverse function is f − 1 ( x ) = 3 − 5 x .
Selecting the Correct Answer Comparing our result with the given options, we see that option D matches our result.
Examples
Imagine you're converting temperatures between Celsius and Fahrenheit. Finding the inverse function is like converting back from Fahrenheit to Celsius. In this case, if f ( x ) converts Celsius to Fahrenheit, then f − 1 ( x ) converts Fahrenheit back to Celsius. This is useful in many real-world applications, such as cooking, weather forecasting, and scientific research, where temperature conversions are frequently needed.
The inverse of the function f ( x ) = 5 3 − x is f − 1 ( x ) = 3 − 5 x . Therefore, the correct answer is option D. This was determined by following the steps to swap x and y and solve for y.
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