QuestionFind $g(f(x))$ for $g(x)=\sqrt{x-1}$ and $f(x)=\sqrt{3-x}$ .

Studdy Solution

STEP 1

Assumptions1. We have two functions, $g(x)=\sqrt{x-1}$ and $f(x)=\sqrt{3-x}$ . . We need to find the composite function $g(f(x))$ .

STEP 2

The composite function $g(f(x))$ is defined as the function $g$ applied to the function $f$ . In other words, wherever we see an $x$ in the function $g$ , we replace it with the function $f(x)$ .
$g(f(x)) = g(\sqrt{-x})$

STEP 3

Now, we substitute the function $f(x)$ into the function $g(x)$ .
$g(f(x)) = \sqrt{\sqrt{3-x}-1}$

STEP 4

implify the expression under the square root.
$g(f(x)) = \sqrt{3-x-1}$

STEP 5

Further simplify the expression under the square root.
$g(f(x)) = \sqrt{2-x}$ The composite function $g(f(x))$ for the given functions $g(x)=\sqrt{x-1}$ and $f(x)=\sqrt{3-x}$ is $g(f(x)) = \sqrt{2-x}$ .

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