Which function is the inverse of [tex]f(x)=-5 x-4[/tex] ?
A. [tex]f^{-1}(x)=-\frac{1}{5} x-\frac{4}{5}[/tex]
B. [tex]f^{-1}(x)=-\frac{1}{5} x+\frac{4}{5}[/tex]
C. [tex]f^{-1}(x)=-4 x+5[/tex]
D. [tex]f^{-1}(x)=4 x+4[/tex]
Answer (2)
Replace f ( x ) with y in the original equation: y = − 5 x − 4 .
Swap x and y to get: x = − 5 y − 4 .
Solve for y in terms of x : y = − 5 1 x − 5 4 .
The inverse function is: f − 1 ( x ) = − 5 1 x − 5 4 .
Explanation
Understanding the Problem The problem asks us to find the inverse of the function f ( x ) = − 5 x − 4 . The inverse function, denoted as f − 1 ( x ) , is a function that 'undoes' the original function. In other words, if we apply f to some value x and then apply f − 1 to the result, we should get back the original value x .
Finding the Inverse Function To find the inverse function, we can follow these steps:
Replace f ( x ) with y : y = − 5 x − 4
Swap x and y : x = − 5 y − 4
Solve for y in terms of x :
Add 4 to both sides: x + 4 = − 5 y
Divide both sides by -5: y = − 5 x + 4
Rewrite the equation: y = − 5 1 x − 5 4
Replace y with f − 1 ( x ) : f − 1 ( x ) = − 5 1 x − 5 4
Verifying the Inverse Function Now, let's verify the result by checking if f ( f − 1 ( x )) = x :
f ( f − 1 ( x )) = − 5 ( − 5 1 x − 5 4 ) − 4 = x + 4 − 4 = x
This confirms that the inverse function we found is correct.
Final Answer Therefore, the inverse function of f ( x ) = − 5 x − 4 is f − 1 ( x ) = − 5 1 x − 5 4 .
Examples
Imagine you're converting temperatures from Celsius to Fahrenheit using a function. The inverse function would then convert Fahrenheit back to Celsius. This concept is useful in many real-world scenarios, such as encoding and decoding messages, or converting between different units of measurement. Understanding inverse functions allows you to reverse a process and return to the original state.
The inverse of the function f ( x ) = − 5 x − 4 is f − 1 ( x ) = − 5 1 x − 5 4 . This was determined through solving the equation by swapping x and y , isolating y , and verifying by substitution. Therefore, the answer is option A.
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