# Math Help - Fixed Point Problem

1. ## Fixed Point Problem

Define $f:[1, \infty ] \rightarrow R$ by $f(x) = 1+ \sqrt{x}$ for all $x \geq 1$. Show that f has exactly one fixed point.

2. You do know that $f$ is strictly increasing on $[1,\infty )$?
The find a solution to $1 + \sqrt t = t$ where $t>1$.

3. Okay, the solution I found was t=1, but you said t > 1. I can't find a solutoin that matches this condition.

4. Originally Posted by tttcomrader
Okay, the solution I found was t=1.
How in the world is t=1 a solution?
$1 + \sqrt 1 = 2 \ne 1.$

5. Originally Posted by tttcomrader
Define $f:[1, \infty ] \rightarrow R$ by $f(x) = 1+ \sqrt{x}$ for all $x \geq 1$. Show that f has exactly one fixed point.
If $x \in [0, \infty)$ is a fixed point of $f$ then:

$
x=f(x)=1+\sqrt{x}
$

Now introduce $y=\sqrt{x}$, solve the equation you get for y, and then sort the roots according to if they correspond to valid solutions of the original fixed point problem.

RonL