Certainly! Let's define and find the composite function \( h \circ g \) (pronounced "h of g"), where \( g \) and \( h \) are given as:
[tex]\[ g = \{(1, 2), (2, 3), (3, 4)\} \][/tex]
[tex]\[ h = \{(2, 3), (3, 4), (4, 5)\} \][/tex]
The composite function \( h \circ g \) is formed by applying \( g \) first and then applying \( h \) to the result of \( g \). Mathematically, for an input \( x \):
[tex]\[ (h \circ g)(x) = h(g(x)) \][/tex]
Let's work through this step-by-step for each input \( x \) in the domain of \( g \):
1. For \( x = 1 \):
- First, find \( g(1) \). From \( g \), we see that \( g(1) = 2 \).
- Next, find \( h(g(1)) = h(2) \). From \( h \), we see that \( h(2) = 3 \).
- Therefore, \( (h \circ g)(1) = 3 \).
2. For \( x = 2 \):
- First, find \( g(2) \). From \( g \), we see that \( g(2) = 3 \).
- Next, find \( h(g(2)) = h(3) \). From \( h \), we see that \( h(3) = 4 \).
- Therefore, \( (h \circ g)(2) = 4 \).
3. For \( x = 3 \):
- First, find \( g(3) \). From \( g \), we see that \( g(3) = 4 \).
- Next, find \( h(g(3)) = h(4) \). From \( h \), we see that \( h(4) = 5 \).
- Therefore, \( (h \circ g)(3) = 5 \).
Combining these results, we get the composite function \( h \circ g \) as:
[tex]\[ h \circ g = \{(1, 3), (2, 4), (3, 5)\} \][/tex]
Now let's represent this composite function in an arrow diagram:
1. From \( 1 \) to \( 3 \).
2. From \( 2 \) to \( 4 \).
3. From \( 3 \) to \( 5 \).
This can be visualized as:
[tex]\[ 1 \xrightarrow{h \circ g} 3 \][/tex]
[tex]\[ 2 \xrightarrow{h \circ g} 4 \][/tex]
[tex]\[ 3 \xrightarrow{h \circ g} 5 \][/tex]
Hence, the composite function \( h \circ g \) is:
[tex]\[ h \circ g = \{(1, 3), (2, 4), (3, 5)\} \][/tex]
This completes the solution.
