Question

Use x^{2} +1" align="absmiddle" class="latex-formula"> with domain [0, ∞] and $g(x) = \sqrt[]{x-4}$ to find each of the following.
1. $[f^{-1}$ ° $g^{-1}](x)$

2. $[f$ ° $g]^{-1} (x)$

Answer 1

Answer:

1. $(f^{-1}og^{-1}) (x) = \sqrt{x^{2}+3 }$

2. $(f o g)^{-1} (x) = x + 3$

Step-by-step explanation:

f(x) = $x^{2}$ + 1

putting f (x) = y

y = $x^{2}$ + 1

$x^{2}$ = ( y-1)

x =  $(y - 1)^{\frac{1}{2} }$

Therefore = $f^{-1}(x)$ = $(x-1)^{\frac{1}{2} }$

g (x) = $\sqrt{x-4}$

putting g(x) = y

y = $\sqrt{x-4}$

$y^{2}$ = (x-4)

x = $y^{2} + 4$

$g^{-1} (x) = x^{2} + 4$

( $f^{-1}$ o $g^{-1}) x$

$(g^{-1}x -1)^{\frac{1}{2} } = \sqrt{x^{2} +3}$

2. (f o g)(x) = (√x-4)²+1) = x-4+1

(f o g)(x) = x-3

Let y = x - 3

x = y + 3

Or $(f o g)^{-1}(x) = x + 3$