Answer :
To solve the problem of evaluating [tex]\((h \circ g)(1)\)[/tex], where [tex]\(g(x) = 2x\)[/tex] and [tex]\(h(x) = x^2 + 4\)[/tex], we'll follow these steps:
1. Determine [tex]\(g(1)\)[/tex]:
Evaluate the function [tex]\(g(x)\)[/tex] at [tex]\(x = 1\)[/tex]:
[tex]\[ g(1) = 2 \cdot 1 = 2 \][/tex]
2. Substitute [tex]\(g(1)\)[/tex] into [tex]\(h(x)\)[/tex]:
Now that we have [tex]\(g(1) = 2\)[/tex], we substitute this result into the function [tex]\(h(x)\)[/tex]:
[tex]\[ h(g(1)) = h(2) \][/tex]
3. Evaluate [tex]\(h(2)\)[/tex]:
Using the function [tex]\(h(x)\)[/tex], evaluate it at [tex]\(x = 2\)[/tex]:
[tex]\[ h(2) = (2)^2 + 4 = 4 + 4 = 8 \][/tex]
Thus, [tex]\((h \circ g)(1) = h(g(1)) = 8\)[/tex].
So the detailed solution gives us:
[tex]\[ (h \circ g)(1) = 8 \][/tex]
1. Determine [tex]\(g(1)\)[/tex]:
Evaluate the function [tex]\(g(x)\)[/tex] at [tex]\(x = 1\)[/tex]:
[tex]\[ g(1) = 2 \cdot 1 = 2 \][/tex]
2. Substitute [tex]\(g(1)\)[/tex] into [tex]\(h(x)\)[/tex]:
Now that we have [tex]\(g(1) = 2\)[/tex], we substitute this result into the function [tex]\(h(x)\)[/tex]:
[tex]\[ h(g(1)) = h(2) \][/tex]
3. Evaluate [tex]\(h(2)\)[/tex]:
Using the function [tex]\(h(x)\)[/tex], evaluate it at [tex]\(x = 2\)[/tex]:
[tex]\[ h(2) = (2)^2 + 4 = 4 + 4 = 8 \][/tex]
Thus, [tex]\((h \circ g)(1) = h(g(1)) = 8\)[/tex].
So the detailed solution gives us:
[tex]\[ (h \circ g)(1) = 8 \][/tex]
