+0

# Piecewise-Defined Functions

0
75
1
+184

For some constants $a$ and $b$ let

$f(x) = \left\{ \begin{array}{cl} 9 - 2x & \text{if } x \le 3, \\ ax + b & \text{if } x > 3. \end{array} \right.$

The function $f$ has the property that $f(f(x)) = x$ for all $x$. What is $a+b$?

Apr 17, 2021

#1
+420
0

if $$f(f(x)) = x$$ for all x, then f(x) is an involution. The properties of an involution include the fact that it is reflexive across the line y=x, which means that y=ax+b must be a reflected version of y=9-2x through the line y=x. To reflect that line, just find the inverse of that line by changing y for x and solving for y:

$$x=9-2y\\2y=9-x\\y=4.5-0.5x$$

a = -0.5, b = 4.5, so a+b $$\boxed{4}$$

Apr 17, 2021