8. If $a(x)=2x^3+x$ And $b(x)=-2x$, What Is \$a(b(2))$[/tex\]?A. -503 B. -132 C. 132 D. $N_{33}$ E. Cannot Be Determined
Understanding Composite Functions
In mathematics, a composite function is a function that is derived from two or more functions. It is a way of combining functions to create a new function. In this article, we will explore how to evaluate composite functions, using the given functions $a(x)=2x^3+x$ and $b(x)=-2x$ as examples.
Evaluating the Inner Function
To evaluate the composite function $a(b(2))$, we first need to find the value of the inner function $b(2)$. The inner function is the function that is evaluated first, and its output is then used as the input for the outer function.
Step 1: Evaluate the Inner Function b(2)
To evaluate the inner function $b(2)$, we substitute $x=2$ into the function $b(x)=-2x$. This gives us:
Step 2: Evaluate the Outer Function a(b(2))
Now that we have the value of the inner function $b(2)=-4$, we can substitute this value into the outer function $a(x)=2x^3+x$. This gives us:
Step 3: Simplify the Expression
To simplify the expression $a(b(2))=2(-4)^3+(-4)$, we first evaluate the exponent $(-4)^3$. This gives us:
Step 4: Substitute the Value of the Exponent
Now that we have the value of the exponent $(-4)^3=-64$, we can substitute this value into the expression $a(b(2))=2(-4)^3+(-4)$. This gives us:
Step 5: Simplify the Expression
To simplify the expression $a(b(2))=2(-64)+(-4)$, we first multiply $2$ by $-64$. This gives us:
Step 6: Add the Remaining Term
Now that we have the value of the first term $2(-64)=-128$, we can add the remaining term $-4$ to get:
Step 7: Simplify the Expression
To simplify the expression $a(b(2))=-128+(-4)$, we first combine the two negative terms. This gives us:
Conclusion
In conclusion, we have evaluated the composite function $a(b(2))$ using the given functions $a(x)=2x^3+x$ and $b(x)=-2x$. We first evaluated the inner function $b(2)=-4$, and then substituted this value into the outer function $a(x)=2x^3+x$. After simplifying the expression, we found that $a(b(2))=-132$.
Final Answer
The final answer is $\boxed{-132}$.
Understanding Composite Functions
In mathematics, a composite function is a function that is derived from two or more functions. It is a way of combining functions to create a new function. In this article, we will explore how to evaluate composite functions, using the given functions $a(x)=2x^3+x$ and $b(x)=-2x$ as examples.
Evaluating the Inner Function
To evaluate the composite function $a(b(2))$, we first need to find the value of the inner function $b(2)$. The inner function is the function that is evaluated first, and its output is then used as the input for the outer function.
Step 1: Evaluate the Inner Function b(2)
To evaluate the inner function $b(2)$, we substitute $x=2$ into the function $b(x)=-2x$. This gives us:
Step 2: Evaluate the Outer Function a(b(2))
Now that we have the value of the inner function $b(2)=-4$, we can substitute this value into the outer function $a(x)=2x^3+x$. This gives us:
Step 3: Simplify the Expression
To simplify the expression $a(b(2))=2(-4)^3+(-4)$, we first evaluate the exponent $(-4)^3$. This gives us:
Step 4: Substitute the Value of the Exponent
Now that we have the value of the exponent $(-4)^3=-64$, we can substitute this value into the expression $a(b(2))=2(-4)^3+(-4)$. This gives us:
Step 5: Simplify the Expression
To simplify the expression $a(b(2))=2(-64)+(-4)$, we first multiply $2$ by $-64$. This gives us:
Step 6: Add the Remaining Term
Now that we have the value of the first term $2(-64)=-128$, we can add the remaining term $-4$ to get:
Step 7: Simplify the Expression
To simplify the expression $a(b(2))=-128+(-4)$, we first combine the two negative terms. This gives us:
Conclusion
In conclusion, we have evaluated the composite function $a(b(2))$ using the given functions $a(x)=2x^3+x$ and $b(x)=-2x$. We first evaluated the inner function $b(2)=-4$, and then substituted this value into the outer function $a(x)=2x^3+x$. After simplifying the expression, we found that $a(b(2))=-132$.
Final Answer
The final answer is $\boxed{-132}$.
Q&A: Evaluating Composite Functions
Q: What is a composite function?
A: A composite function is a function that is derived from two or more functions. It is a way of combining functions to create a new function.
Q: How do I evaluate a composite function?
A: To evaluate a composite function, you need to first evaluate the inner function, and then substitute the value of the inner function into the outer function.
Q: What is the inner function?
A: The inner function is the function that is evaluated first, and its output is then used as the input for the outer function.
Q: What is the outer function?
A: The outer function is the function that is evaluated second, and it takes the output of the inner function as its input.
Q: How do I simplify a composite function?
A: To simplify a composite function, you need to follow the order of operations (PEMDAS) and evaluate the expressions inside the parentheses first.
Q: What is the final answer to the problem $a(b(2))$?
A: The final answer to the problem $a(b(2))$ is $-132$.
Q: Can I use a calculator to evaluate a composite function?
A: Yes, you can use a calculator to evaluate a composite function. However, it is always a good idea to check your work by hand to make sure you get the correct answer.
Q: What if I get a different answer when using a calculator?
A: If you get a different answer when using a calculator, it may be due to a mistake in your calculation. Double-check your work and make sure you are following the correct order of operations.
Q: Can I use a composite function to solve a real-world problem?
A: Yes, composite functions can be used to solve a wide range of real-world problems. For example, you can use a composite function to model the growth of a population or the cost of a product.
Q: How do I know if a composite function is a function?
A: A composite function is a function if it passes the vertical line test. This means that for every input, there is only one output.
Q: Can I use a composite function to solve a problem that involves multiple variables?
A: Yes, you can use a composite function to solve a problem that involves multiple variables. However, you need to make sure that the composite function is well-defined and that it takes into account all the variables involved.
Q: What is the difference between a composite function and a function of a function?
A: A composite function is a function that is derived from two or more functions, whereas a function of a function is a function that takes another function as its input.