+54 1.)
Given that \[ f(x) = \dfrac {1}{1 - \dfrac {1}{1 - \dfrac {1}{1 - x}}}, \] compute $(f(f( - 2)))^{ - 2}$. Express your answer as a common fraction.
2.)
For positive integers $a$, $b$, and $c$, what is the value of the product $abc$? \[ \dfrac {1}{a + \dfrac {1}{b + \dfrac {1}{c}}} = \dfrac38 \]\
3.)
What is the value of the following expression? \[ \sqrt {6 + \sqrt {6 + \sqrt {6 + \cdots}}} \]
4.)
Fully simplify $\sqrt {14 + 8\sqrt {3}}$.
5.)
Suppose $f(x)=x+1$. For what value of $x$ is $\overbrace{f(f(\cdots(f}^\text{2015 total \textit{f}s}(x))\cdots)=0$?
+128474 3)
Let x = √[ 6 + √ [6 + √6 +.... ] ] square both sides
x^2 = 6 + √[ 6 + √ [6 + √6 +.... ] ]
x^2 = 6 + x rearrange as
x^2 - x - 6 = 0 factor
(x - 3) ( x + 2) = 0
Setting each factor to 0 and solving for x gives that x = 3 or x = -2
Reject the second
So.....x = 3 = the value
+128474 \( $\sqrt {14 + 8\sqrt {3}}$\)
In the form √ [ a + b√c ] we can find a simplification if
(b/2)^2 + ( √c ) ^2 = a
And the simpification becomes √ [ ( (b/2) + √c )^2 ]
(4)^2 + 3 = 19 isn't true.....but notice that we can express this as
√ [ 2 ( 7 + 4√3) ] = √2 * √ [ 7 + 4√3 ]
Then considering the second radical....
(4/2)^2 + (√3)^2 = 2^2 + 3 = 7 is true
So we can write
√2 * √ [ ( (4/2) + √3 ) ^2 ] =
√ 2 * √ [ ( 2 + √3)^2 ] =
√2 [ 2 + √3] =
2√2 + √6 =
√8 + √6
