Identifying the Function Rule

Oct 13, 2025 by miodeya.com 133 views

Understanding the Problem

The problem presents a simple mathematical relationship between the input and output of a function. We are given that the output is one more than twice the input, and we need to write a function rule to represent this relationship. The input is denoted by the variable $$x$$, and the output is denoted by the variable $$y$$.

Identifying the Function Rule

To write the function rule, we need to identify the relationship between the input and output. The problem states that the output is one more than twice the input, which can be represented as $$y = 2x + 1$$. This equation states that the output $$y$$ is equal to twice the input $$x$$, plus one.

Analyzing the Function Rule

Let's analyze the function rule $$y = 2x + 1$$ to understand its behavior. The function rule consists of two parts: the input $$x$$ and the output $$y$$. The input $$x$$ is multiplied by 2, which means that the output will be twice the input. Then, 1 is added to the result, which means that the output will be one more than twice the input.

Graphing the Function Rule

To visualize the function rule, we can graph the equation $$y = 2x + 1$$. The graph will be a straight line with a slope of 2 and a y-intercept of 1. The graph will show that as the input $$x$$ increases, the output $$y$$ also increases, but at a rate of twice the input.

Real-World Applications

The function rule $$y = 2x + 1$$ has many real-world applications. For example, if we are selling tickets to a concert, the price of the ticket may be twice the cost of the ticket, plus a service charge of $1. In this case, the function rule can be used to calculate the price of the ticket based on the cost of the ticket.

Solving for the Input

To solve for the input $$x$$, we can use the function rule $$y = 2x + 1$$ and isolate the variable $$x$$. We can do this by subtracting 1 from both sides of the equation, which gives us $$y - 1 = 2x$$. Then, we can divide both sides of the equation by 2, which gives us $$x = \frac{y - 1}{2}$$.

Solving for the Output

To solve for the output $$y$$, we can use the function rule $$y = 2x + 1$$ and isolate the variable $$y$$. We can do this by subtracting 2x from both sides of the equation, which gives us $$y - 2x = 1$$. Then, we can add 2x to both sides of the equation, which gives us $$y = 2x + 1$$.

Conclusion

In conclusion, the function rule $$y = 2x + 1$$ represents the relationship between the input and output of a function. The function rule states that the output is one more than twice the input, and it can be used to calculate the output based on the input. The function rule has many real-world applications, and it can be used to solve for the input and output.

Example Problems

Problem 1

If the input $$x$$ is 3, what is the output $$y$$?

Solution

To solve for the output $$y$$, we can use the function rule $$y = 2x + 1$$ and substitute the value of $$x$$ into the equation. We get $$y = 2(3) + 1 = 6 + 1 = 7$$.

Problem 2

If the output $$y$$ is 9, what is the input $$x$$?

Solution

Practice Problems

Problem 1

If the input $$x$$ is 5, what is the output $$y$$?

Problem 2

If the output $$y$$ is 11, what is the input $$x$$?

Problem 3

If the input $$x$$ is 2, what is the output $$y$$?

Problem 4

If the output $$y$$ is 15, what is the input $$x$$?

Problem 5

If the input $$x$$ is 1, what is the output $$y$$?

Problem 6

If the output $$y$$ is 7, what is the input $$x$$?

Problem 7

If the input $$x$$ is 4, what is the output $$y$$?

Problem 8

If the output $$y$$ is 13, what is the input $$x$$?

Problem 9

If the input $$x$$ is 3, what is the output $$y$$?

Problem 10

If the output $$y$$ is 9, what is the input $$x$$?

Answer Key

Problem 1

The output $$y$$ is 11.

Problem 2

The input $$x$$ is 5.

Problem 3

The output $$y$$ is 5.

Problem 4

The input $$x$$ is 7.

Problem 5

The output $$y$$ is 3.

Problem 6

The input $$x$$ is 3.

Problem 7

The output $$y$$ is 9.

Problem 8

The input $$x$$ is 6.

Problem 9

The output $$y$$ is 7.

Problem 10

The input $$x$$ is 4.

Understanding the Problem

Q&A

Q: What is the function rule for the output being one more than twice the input?

A: The function rule is $$y = 2x + 1$$.

Q: How do I graph the function rule?

A: To graph the function rule, we can use the equation $$y = 2x + 1$$. The graph will be a straight line with a slope of 2 and a y-intercept of 1.

Q: What is the real-world application of the function rule?

A: The function rule has many real-world applications. For example, if we are selling tickets to a concert, the price of the ticket may be twice the cost of the ticket, plus a service charge of $1.

Q: How do I solve for the input $$x$$?

A: To solve for the input $$x$$, we can use the function rule $$y = 2x + 1$$ and isolate the variable $$x$$. We can do this by subtracting 1 from both sides of the equation, which gives us $$y - 1 = 2x$$. Then, we can divide both sides of the equation by 2, which gives us $$x = \frac{y - 1}{2}$$.

Q: How do I solve for the output $$y$$?

A: To solve for the output $$y$$, we can use the function rule $$y = 2x + 1$$ and isolate the variable $$y$$. We can do this by subtracting 2x from both sides of the equation, which gives us $$y - 2x = 1$$. Then, we can add 2x to both sides of the equation, which gives us $$y = 2x + 1$$.

Q: What are some example problems that I can use to practice the function rule?

A: Here are some example problems that you can use to practice the function rule:

If the input $$x$$ is 3, what is the output $$y$$?
If the output $$y$$ is 9, what is the input $$x$$?
If the input $$x$$ is 5, what is the output $$y$$?
If the output $$y$$ is 11, what is the input $$x$$?

Q: What are some practice problems that I can use to reinforce my understanding of the function rule?

A: Here are some practice problems that you can use to reinforce your understanding of the function rule:

If the input $$x$$ is 2, what is the output $$y$$?
If the output $$y$$ is 7, what is the input $$x$$?
If the input $$x$$ is 4, what is the output $$y$$?
If the output $$y$$ is 13, what is the input $$x$$?

Answer Key

Problem 1

The output $$y$$ is 11.

Problem 2

The input $$x$$ is 5.

Problem 3

The output $$y$$ is 5.

Problem 4

The input $$x$$ is 7.

Problem 5

The output $$y$$ is 3.

Problem 6

The input $$x$$ is 3.

Problem 7

The output $$y$$ is 9.

Problem 8

The input $$x$$ is 6.

Problem 9

The output $$y$$ is 7.

Problem 10

The input $$x$$ is 4.