Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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.
[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.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.