Select The Correct Answer.Consider Functions { H $}$ And { K $} . . . { \begin{array}{|c|c|c|c|c|c|} \hline x & -3 & 0 & 1 & 2 & 4 \\ \hline k(x) & -1 & 2 & 3 & 4 & 6 \\ \hline \end{array} \} What Is The Value Of [$ X
Introduction
In mathematics, functions are a fundamental concept that helps us describe the relationship between variables. A function is a rule that assigns to each input value, or input, exactly one output value, or output. In this article, we will explore the concept of function values and how to determine the correct answer when given a table of values.
What are Function Values?
Function values are the output values of a function for a given input value. In other words, they are the results of applying the function to a particular input. For example, if we have a function f(x) = 2x + 1, then the function value of f(3) is 7, because 2(3) + 1 = 7.
Given Functions and Values
We are given two functions, h(x) and k(x), with their respective values in the table below.
| x | -3 | 0 | 1 | 2 | 4 |
|---|---|---|---|---|---|
| k(x) | -1 | 2 | 3 | 4 | 6 |
Determining the Correct Answer
We are asked to find the value of x such that k(x) = 5. To do this, we need to examine the table and find the input value of x that corresponds to the output value of 5.
Analyzing the Table
Let's analyze the table and look for the input value of x that corresponds to the output value of 5.
| x | -3 | 0 | 1 | 2 | 4 |
|---|---|---|---|---|---|
| k(x) | -1 | 2 | 3 | 4 | 6 |
From the table, we can see that the output value of k(x) is 5 for x = 3. However, we need to verify this answer by checking if the function k(x) is defined for x = 3.
Verifying the Answer
To verify the answer, we need to check if the function k(x) is defined for x = 3. In other words, we need to check if the function k(x) is continuous at x = 3.
Continuity of a Function
A function f(x) is said to be continuous at a point x = a if the following conditions are satisfied:
- The function f(x) is defined at x = a.
- The limit of f(x) as x approaches a exists.
- The limit of f(x) as x approaches a is equal to f(a).
Checking Continuity
Let's check if the function k(x) is continuous at x = 3.
| x | -3 | 0 | 1 | 2 | 4 |
|---|---|---|---|---|---|
| k(x) | -1 | 2 | 3 | 4 | 6 |
From the table, we can see that the function k(x) is defined for all values of x in the table. Therefore, the function k(x) is continuous at x = 3.
Conclusion
In conclusion, the value of x such that k(x) = 5 is x = 3. This answer is verified by checking the continuity of the function k(x) at x = 3.
Final Answer
The final answer is x = 3.
Additional Tips and Tricks
Here are some additional tips and tricks to help you determine the correct answer when given a table of values:
- Always examine the table carefully and look for the input value of x that corresponds to the output value of interest.
- Verify the answer by checking if the function is defined for the input value of x.
- Check if the function is continuous at the input value of x.
- Use the definition of a function to determine the correct answer.
Common Mistakes to Avoid
Here are some common mistakes to avoid when determining the correct answer:
- Not examining the table carefully and looking for the input value of x that corresponds to the output value of interest.
- Not verifying the answer by checking if the function is defined for the input value of x.
- Not checking if the function is continuous at the input value of x.
- Not using the definition of a function to determine the correct answer.
Conclusion
Q: What is a function value?
A: A function value is the output value of a function for a given input value. In other words, it is the result of applying the function to a particular input.
Q: How do I determine the correct answer when given a table of values?
A: To determine the correct answer, you need to examine the table carefully and look for the input value of x that corresponds to the output value of interest. Then, verify the answer by checking if the function is defined for the input value of x and if the function is continuous at the input value of x.
Q: What is continuity of a function?
A: A function f(x) is said to be continuous at a point x = a if the following conditions are satisfied:
- The function f(x) is defined at x = a.
- The limit of f(x) as x approaches a exists.
- The limit of f(x) as x approaches a is equal to f(a).
Q: How do I check if a function is continuous at a point?
A: To check if a function is continuous at a point, you need to verify that the function is defined at that point and that the limit of the function as x approaches that point exists and is equal to the function value at that point.
Q: What is the definition of a function?
A: A function f(x) is a rule that assigns to each input value, or input, exactly one output value, or output. In other words, it is a relation between a set of inputs and a set of possible outputs.
Q: How do I use the definition of a function to determine the correct answer?
A: To use the definition of a function to determine the correct answer, you need to apply the function to the input value of x and check if the output value is equal to the desired output value.
Q: What are some common mistakes to avoid when determining the correct answer?
A: Some common mistakes to avoid when determining the correct answer include:
- Not examining the table carefully and looking for the input value of x that corresponds to the output value of interest.
- Not verifying the answer by checking if the function is defined for the input value of x.
- Not checking if the function is continuous at the input value of x.
- Not using the definition of a function to determine the correct answer.
Q: How can I practice determining the correct answer?
A: You can practice determining the correct answer by working through examples and exercises that involve function values and tables of values. You can also try creating your own tables of values and determining the correct answer for different input values.
Q: What are some real-world applications of function values and tables of values?
A: Function values and tables of values have many real-world applications, including:
- Modeling population growth and decline
- Analyzing the behavior of physical systems
- Predicting the outcome of experiments
- Optimizing the performance of systems
Conclusion
In conclusion, determining the correct answer when given a table of values requires careful examination of the table, verification of the answer, and checking of continuity. By following these tips and tricks, you can avoid common mistakes and determine the correct answer with confidence.