Questions: For the real-valued functions (g(x)=2 x+5) and (h(x)=sqrtx-2), find the composition (g circ h) and specify its domain using interval notation. ((g circ h)(x)=) Domain of (g circ h) :

For the real-valued functions (g(x)=2 x+5) and (h(x)=sqrtx-2), find the composition (g circ h) and specify its domain using interval notation.

((g circ h)(x)=)

Domain of (g circ h) :

Transcript text: For the real-valued functions $g(x)=2 x+5$ and $h(x)=\sqrt{x-2}$, find the composition $g \circ h$ and specify its domain using interval notation. \[ (g \circ h)(x)= \] Domain of $g \circ h$ :

Solution

Solution Steps

To find the composition of the functions $ g(x) = 2x + 5 $ and $ h(x) = \sqrt{x - 2} $, we need to substitute $ h(x) $ into $ g(x) $. This means we will evaluate $ g(h(x)) $. Additionally, we need to determine the domain of the composition function, which is the set of all $ x $ values for which $ h(x) $ is defined and $ g(h(x)) $ is also defined.

Composition: Substitute $ h(x) $ into $ g(x) $.
Domain: Determine the domain of $ h(x) $ and ensure that the output of $ h(x) $ is within the domain of $ g(x) $.

Step 1: Define the Functions

Given the functions: \[ g(x) = 2x + 5 \] \[ h(x) = \sqrt{x - 2} \]

Step 2: Find the Composition $ g \circ h $

To find the composition $ (g \circ h)(x) $, substitute $ h(x) $ into $ g(x) $: \[ (g \circ h)(x) = g(h(x)) = g(\sqrt{x - 2}) \] \[ g(\sqrt{x - 2}) = 2\sqrt{x - 2} + 5 \]

Step 3: Determine the Domain of $ g \circ h $

The domain of $ g \circ h $ is determined by the domain of $ h(x) $ and ensuring that the output of $ h(x) $ is within the domain of $ g(x) $.

The domain of $ h(x) = \sqrt{x - 2} $ requires: \[ x - 2 \geq 0 \] \[ x \geq 2 \]
Since $ g(x) = 2x + 5 $ is defined for all real numbers, the domain of $ g \circ h $ is the same as the domain of $ h(x) $: \[ x \geq 2 \]

In interval notation, the domain is: \[ [2, \infty) \]