$\begin{array}{l}
f(x)=\frac{x}{2}+2 \text { for } x \in R \\
g(x)=x^2-2 x \text { for } x \in R
\end{array}$
Solve the equation: $f(g(x))=f^{-1}(x)$
Answer (2)
To solve f ( g ( x )) = f − 1 ( x ) , we computed f ( g ( x )) and found the inverse of f ( x ) . This led to the quadratic equation x 2 − 6 x + 12 = 0 , which has complex solutions: x = 3 ± i 3 .
;
Find the inverse function: f − 1 ( x ) = 2 x − 4 .
Compute the composite function: f ( g ( x )) = f r a c x 2 − 2 x + 4 2 .
Set up the equation: f r a c x 2 − 2 x + 4 2 = 2 x − 4 .
Solve the quadratic equation: x = 3 p mi s q r t 3 .
Explanation
Analyze the functions First, let's analyze the given functions. We have f ( x ) = 2 x + 2 and g ( x ) = x 2 − 2 x . We need to solve the equation f ( g ( x )) = f − 1 ( x ) .
Find the inverse function Next, we need to find the inverse function f − 1 ( x ) . To do this, we set y = f ( x ) = 2 x + 2 and solve for x in terms of y . So, y = 2 x + 2 ⟹ y − 2 = 2 x ⟹ x = 2 ( y − 2 ) = 2 y − 4 . Therefore, f − 1 ( x ) = 2 x − 4 .
Compute f(g(x)) Now, we need to compute f ( g ( x )) . Since f ( x ) = 2 x + 2 and g ( x ) = x 2 − 2 x , we have f ( g ( x )) = 2 g ( x ) + 2 = 2 x 2 − 2 x + 2 = 2 x 2 − 2 x + 4 .
Set up the equation Now we set up the equation f ( g ( x )) = f − 1 ( x ) : 2 x 2 − 2 x + 4 = 2 x − 4 .
Solve the equation Next, we solve the equation for x : x 2 − 2 x + 4 = 4 x − 8 ⟹ x 2 − 6 x + 12 = 0 .
Use the quadratic formula Now, we use the quadratic formula to find the roots of the quadratic equation x 2 − 6 x + 12 = 0 . The quadratic formula is x = 2 a − b ± b 2 − 4 a c . In this case, a = 1 , b = − 6 , and c = 12 . So, x = 2 ( 1 ) 6 ± ( − 6 ) 2 − 4 ( 1 ) ( 12 ) = 2 6 ± 36 − 48 = 2 6 ± − 12 = 2 6 ± 2 i 3 = 3 ± i 3 .
Final solutions Therefore, the solutions are x = 3 + i 3 and x = 3 − i 3 .
Examples
Understanding function composition and inverse functions is crucial in many areas of mathematics and engineering. For example, in cryptography, functions and their inverses are used to encrypt and decrypt messages. In computer graphics, transformations such as scaling, rotation, and translation can be represented as functions, and their inverses are used to undo these transformations. Also, in control systems, understanding the inverse of a system's transfer function is essential for designing controllers that can accurately track desired outputs. The ability to manipulate and solve equations involving functions and their inverses is a fundamental skill that has wide-ranging applications.
