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

The value of a+b is 4.

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=-(1)/(2)`

Substitute this value in equation (1).

`9(-(1)/(2))+b=0`

`b=(9)/(2)`

The value of a is
`-(1)/(2)` and value of b is
`(9)/(2)`. The value of a+b is

`a+b=-(1)/(2)+(9)/(2)`

`a+b=4`

Therefore the value of a+b is 4.