To solve for [tex]\((g \circ h)(4)\)[/tex], we need to understand what the notation [tex]\((g \circ h)\)[/tex] indicates. The notation [tex]\((g \circ h)(x)\)[/tex] refers to the composition of the functions [tex]\(g\)[/tex] and [tex]\(h\)[/tex], meaning that we first apply the function [tex]\(h\)[/tex] to [tex]\(x\)[/tex], and then apply the function [tex]\(g\)[/tex] to the result of [tex]\(h(x)\)[/tex].
Given the functions:
[tex]\[ g(x) = 5x - 3 \][/tex]
[tex]\[ h(x) = \sqrt{x} \][/tex]
We want to find [tex]\((g \circ h)(4)\)[/tex], which means:
[tex]\[ (g \circ h)(4) = g(h(4)) \][/tex]
First, let's evaluate [tex]\(h(4)\)[/tex]:
[tex]\[ h(4) = \sqrt{4} = 2 \][/tex]
Next, we take this result and apply the function [tex]\(g\)[/tex] to it:
[tex]\[ g(h(4)) = g(2) \][/tex]
Now, let's find [tex]\(g(2)\)[/tex]:
[tex]\[ g(2) = 5 \cdot 2 - 3 = 10 - 3 = 7 \][/tex]
Therefore, [tex]\((g \circ h)(4)\)[/tex] is:
[tex]\[ (g \circ h)(4) = 7 \][/tex]
In summary:
[tex]\[ h(4) = 2 \][/tex]
[tex]\[ g(h(4)) = g(2) = 7 \][/tex]
So, the final answer is:
[tex]\[ (g \circ h)(4) = 7 \][/tex]
