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Graphs and Functions composition of two functions Advanced For the real-valued functions $f(x)=4x+3$ and $g(x)=\sqrt{x-3}$ , find the composition $f\circ g$ and specify its domain using interval notation.

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Domain and range of square root functions: equations

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Q. Graphs and Functions composition of two functions Advanced For the real-valued functions

f(x)=4x+3

and

g(x)=\sqrt{x-3}

, find the composition

f\circ g

and specify its domain using interval notation.

Identify functions: Identify the functions to be composed. $f(x) = 4x + 3$ and $g(x) = \sqrt{x - 3}$ .
Compose f and g: Compose f and g to find $f(g(x))$ . $\newline$ $f(g(x)) = f(\sqrt{x - 3}) = 4(\sqrt{x - 3}) + 3$ .
Determine domain of $g$ : Determine the domain of $g(x)$ , since it affects the domain of $f(g(x))$ . $g(x) = \sqrt{x - 3}$ is defined for $x - 3 \geq 0$ , which simplifies to $x \geq 3$ .
Check restrictions: Check if there are any additional restrictions from $f(x)$ that could affect the domain of $f(g(x))$ . $f(x) = 4x + 3$ does not impose any further restrictions since it is defined for all real $x$ .
Combine domain restrictions: Combine the domain restrictions. The domain of $f(g(x))$ is $x \geq 3$ .

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