Let
[tex]
\begin{array}{l}
f(x)=\frac{x+2}{x+7} \\
f^{-1}(-2)=\square
\end{array}
[/tex]
Answer (1)
Set f ( x ) = − 2 , which gives x + 7 x + 2 = − 2 .
Solve for x : x + 2 = − 2 ( x + 7 ) .
Simplify the equation: x + 2 = − 2 x − 14 .
Solve for x : 3 x = − 16 , so x = − 3 16 .
Therefore, f − 1 ( − 2 ) = − 3 16 .
Explanation
Understanding the Problem We are given the function f ( x ) = x + 7 x + 2 and asked to find f − 1 ( − 2 ) . This means we need to find the value x such that f ( x ) = − 2 .
Setting up the Equation To find f − 1 ( − 2 ) , we set f ( x ) = − 2 and solve for x . So we have x + 7 x + 2 = − 2.
Solving for x Now, we solve for x . Multiply both sides by ( x + 7 ) to get x + 2 = − 2 ( x + 7 ) x + 2 = − 2 x − 14 Add 2 x to both sides: 3 x + 2 = − 14 Subtract 2 from both sides: 3 x = − 16 Divide by 3: x = − 3 16
Finding the Inverse Value Therefore, f − 1 ( − 2 ) = − 3 16 .
Examples
Imagine you are tuning a radio and the frequency is given by the function f ( x ) = x + 7 x + 2 . If you want to find the input x that gives a specific frequency of -2, you need to find the inverse function f − 1 ( − 2 ) . This is similar to finding the exact setting on the radio dial to get the desired station. Understanding inverse functions helps in many practical applications where you need to reverse a process or find the input that produces a specific output.
