If $h(x)=x^3+x$ And $g(x)=2x+3$, Then \$g(h(2))=$[/tex\]?F. $7$ G. $10$ H. \$17$[/tex\] I. $19$ J. $23$
Understanding the Problem
To solve this problem, we need to understand the composition of functions and how to evaluate them. The given functions are $h(x)=x^3+x$ and $g(x)=2x+3$. We are asked to find the value of $g(h(2))$.
Evaluating the Inner Function
The first step is to evaluate the inner function, which is $h(2)$. To do this, we substitute $x=2$ into the function $h(x)=x^3+x$. This gives us:
Evaluating the Outer Function
Now that we have the value of $h(2)$, we can substitute it into the outer function, which is $g(x)=2x+3$. This gives us:
Conclusion
Therefore, the value of $g(h(2))$ is $23$.
Understanding the Composition of Functions
The composition of functions is a way of combining two or more functions to create a new function. In this case, we have two functions, $h(x)$ and $g(x)$, and we want to find the value of $g(h(2))$. To do this, we need to evaluate the inner function, $h(2)$, and then substitute the result into the outer function, $g(x)$.
Evaluating the Composition of Functions
To evaluate the composition of functions, we need to follow the order of operations. First, we evaluate the inner function, and then we substitute the result into the outer function. In this case, we first evaluate $h(2)$, and then we substitute the result into $g(x)$.
Real-World Applications
The composition of functions has many real-world applications. For example, in physics, the composition of functions is used to model the motion of objects. In economics, the composition of functions is used to model the behavior of markets. In computer science, the composition of functions is used to model the behavior of algorithms.
Conclusion
In conclusion, the value of $g(h(2))$ is $23$. The composition of functions is a powerful tool that can be used to model complex systems and behaviors. By understanding how to evaluate the composition of functions, we can gain a deeper understanding of the world around us.
Frequently Asked Questions
- Q: What is the composition of functions? A: The composition of functions is a way of combining two or more functions to create a new function.
- Q: How do I evaluate the composition of functions? A: To evaluate the composition of functions, you need to follow the order of operations. First, evaluate the inner function, and then substitute the result into the outer function.
- Q: What are some real-world applications of the composition of functions? A: The composition of functions has many real-world applications, including modeling the motion of objects in physics, modeling the behavior of markets in economics, and modeling the behavior of algorithms in computer science.
Final Answer
The final answer is $23$.
Understanding the Composition of Functions
The composition of functions is a way of combining two or more functions to create a new function. This is a fundamental concept in mathematics and has many real-world applications.
Frequently Asked Questions
Q: What is the composition of functions?
A: The composition of functions is a way of combining two or more functions to create a new function. This is denoted by the notation $f(g(x))$, where $f(x)$ and $g(x)$ are the two functions being composed.
Q: How do I evaluate the composition of functions?
A: To evaluate the composition of functions, you need to follow the order of operations. First, evaluate the inner function, and then substitute the result into the outer function. For example, if we have $f(x)=x^2+1$ and $g(x)=2x+3$, then $f(g(x))$ would be evaluated as follows:
- Evaluate the inner function: $g(x)=2x+3$
- Substitute the result into the outer function: $f(g(x))=f(2x+3)=(2x+3)^2+1$
Q: What are some real-world applications of the composition of functions?
A: The composition of functions has many real-world applications, including:
- Modeling the motion of objects in physics
- Modeling the behavior of markets in economics
- Modeling the behavior of algorithms in computer science
- Modeling the behavior of complex systems in engineering
Q: How do I know if a function is composite?
A: A function is composite if it can be expressed as the composition of two or more functions. For example, the function $f(x)=x^2+1$ is not composite, but the function $f(x)=(2x+3)^2+1$ is composite because it can be expressed as the composition of the functions $g(x)=2x+3$ and $f(x)=x^2+1$.
Q: Can a function be both composite and non-composite?
A: No, a function cannot be both composite and non-composite. If a function is composite, then it can be expressed as the composition of two or more functions. If a function is not composite, then it cannot be expressed as the composition of two or more functions.
Q: How do I determine if a function is composite or non-composite?
A: To determine if a function is composite or non-composite, you need to check if it can be expressed as the composition of two or more functions. If it can, then it is composite. If it cannot, then it is non-composite.
Q: What are some common types of composite functions?
A: Some common types of composite functions include:
- Function composition: $f(g(x))$
- Function iteration: $f(f(x))$
- Function recursion: $f(f(f(x)))$
Q: How do I evaluate a composite function?
A: To evaluate a composite function, you need to follow the order of operations. First, evaluate the inner function, and then substitute the result into the outer function.
Q: Can a composite function have multiple inputs?
A: Yes, a composite function can have multiple inputs. For example, the function $f(x,y)=x2+y2$ is a composite function because it can be expressed as the composition of the functions $g(x)=x^2$ and $h(y)=y^2$.
Q: How do I determine if a composite function is invertible?
A: To determine if a composite function is invertible, you need to check if it has a unique inverse. If it does, then it is invertible. If it does not, then it is not invertible.
Q: What are some common applications of composite functions in real-world scenarios?
A: Some common applications of composite functions in real-world scenarios include:
- Modeling the motion of objects in physics
- Modeling the behavior of markets in economics
- Modeling the behavior of algorithms in computer science
- Modeling the behavior of complex systems in engineering
Conclusion
In conclusion, the composition of functions is a powerful tool that can be used to model complex systems and behaviors. By understanding how to evaluate the composition of functions, we can gain a deeper understanding of the world around us.