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) :

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.

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

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

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

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 $g(x)$.

1. The domain of $h(x) = \sqrt{x - 2}$ requires: $$x - 2 \geq 0$$ $$x \geq 2$$

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: $$[2, \infty)$$

Final Answer