Question

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?

Answer 1

Answer:

The value of a+b is 4.

Step-by-step explanation:

The given function is

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

It is given that for some constants a and b the function f has the property that f(f(x))=x for all x.

For x≤3,

$f(x)=9-2x$

For x>3,

$f(x)=ax+b$

At x=0,

$f(0)=9-2(0)=9$

$f(f(0))=f(9)\Rightarrow a(9)+b=9a+b$

Using property f(f(x))=x,

$f(f(0))=0$

$9a+b=0$                     .... (1)

At x=1,

$f(1)=9-2(1)=7$

$f(f(1))=f(7)\Rightarrow a(7)+b=7a+b$

Using property f(f(x))=x,

$f(f(1))=1$

$7a+b=1$                     .... (2)

Subtract equation (2) from equation (1).

$9a+b-(7a+b)=0-1$    

$2a=-1$

Divide both sides by 2.

$a=-\frac{1}{2}$

Substitute this value in equation (1).

$9(-\frac{1}{2})+b=0$

$b=\frac{9}{2}$

The value of a is $-\frac{1}{2}$ and value of b is $\frac{9}{2}$. The value of a+b is

$a+b=-\frac{1}{2}+\frac{9}{2}$

$a+b=4$

Therefore the value of a+b is 4.