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In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). It is a way of describing a relationship between variables, where each input corresponds to exactly one output. In this article, we will explore which of the given equations represent functions and why.

What is a Function?

A function is a relation between a set of inputs (domain) and a set of possible outputs (range). It is a way of describing a relationship between variables, where each input corresponds to exactly one output. In other words, for every input, there is only one output. This is a fundamental concept in mathematics, and it is essential to understand what constitutes a function.

Equation A: $y = 4x + 13$

This equation represents a linear function. It is in the form of $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. In this case, the slope is $4$ and the y-intercept is $13$ . This equation represents a straight line, and for every input $x$ , there is only one output $y$ . Therefore, this equation represents a function.

Equation B: $x = 5$

This equation does not represent a function. It is an equation that sets $x$ equal to a constant value, $5$ . This means that there is only one possible input, and there is no corresponding output. In other words, this equation does not describe a relationship between variables, and it does not meet the definition of a function.

Equation C: $x^2 + y^2 = 16$

This equation represents a circle with a radius of $4$ . It is not a function because it does not pass the vertical line test. In other words, for a given value of $x$ , there are multiple possible values of $y$ . This means that this equation does not meet the definition of a function.

Equation D: $y^2 = \frac{1}{3}x - 6$

This equation represents a parabola. However, it is not a function because it does not pass the vertical line test. In other words, for a given value of $x$ , there are multiple possible values of $y$ . This means that this equation does not meet the definition of a function.

Equation E: $y = 3x^2 - x - 1$

This equation represents a quadratic function. It is in the form of $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. In this case, $a = 3$ , $b = -1$ , and $c = -1$ . This equation represents a parabola, and for every input $x$ , there is only one output $y$ . Therefore, this equation represents a function.

Conclusion

In conclusion, the equations that represent functions are:

Equation A: $y = 4x + 13$
Equation E: $y = 3x^2 - x - 1$

The equations that do not represent functions are:

Equation B: $x = 5$
Equation C: $x^2 + y^2 = 16$
Equation D: $y^2 = \frac{1}{3}x - 6$

It is essential to understand what constitutes a function and how to identify functions in mathematical equations. This knowledge is crucial in mathematics, science, and engineering, and it has numerous applications in real-world problems.

Key Takeaways

A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
A function must pass the vertical line test, meaning that for every input, there is only one output.
Equations that represent functions are in the form of $y = f(x)$ , where $f(x)$ is a function of $x$ .
Equations that do not represent functions are those that do not pass the vertical line test or are not in the form of $y = f(x)$ .

Real-World Applications

Understanding functions is essential in various fields, including:

Science: Functions are used to model real-world phenomena, such as population growth, chemical reactions, and physical systems.
Engineering: Functions are used to design and optimize systems, such as electronic circuits, mechanical systems, and computer algorithms.
Economics: Functions are used to model economic systems, such as supply and demand curves, and to make predictions about economic outcomes.

In our previous article, we explored the concept of functions and identified which of the given equations represent functions. In this article, we will provide a Q&A guide to help you better understand functions and how to identify them.

Q: What is a function?

A: A function is a relation between a set of inputs (domain) and a set of possible outputs (range). It is a way of describing a relationship between variables, where each input corresponds to exactly one output.

Q: What are the characteristics of a function?

A: A function must have the following characteristics:

Domain: A set of possible inputs.
Range: A set of possible outputs.
Mapping: Each input corresponds to exactly one output.
No repeated outputs: For every input, there is only one output.

Q: How do I identify a function?

A: To identify a function, you can use the following steps:

Check the domain: Make sure the equation has a well-defined domain.
Check the range: Make sure the equation has a well-defined range.
Check the mapping: Make sure each input corresponds to exactly one output.
Check for repeated outputs: Make sure there are no repeated outputs for any input.

Q: What are some common types of functions?

A: Some common types of functions include:

Linear functions: Equations in the form of $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
Quadratic functions: Equations in the form of $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.
Polynomial functions: Equations in the form of $y = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$ , where $a_n$ , $a_{n-1}$ , $\ldots$ , $a_1$ , and $a_0$ are constants.
Rational functions: Equations in the form of $y = \frac{f(x)}{g(x)}$ , where $f(x)$ and $g(x)$ are polynomials.

Q: How do I graph a function?

A: To graph a function, you can use the following steps:

Determine the domain: Identify the set of possible inputs.
Determine the range: Identify the set of possible outputs.
Plot points: Plot points on the graph that correspond to the inputs and outputs.
Draw the graph: Draw the graph of the function using the plotted points.

Q: What are some common mistakes to avoid when identifying functions?

A: Some common mistakes to avoid when identifying functions include:

Not checking the domain: Failing to check the domain of the equation can lead to incorrect conclusions.
Not checking the range: Failing to check the range of the equation can lead to incorrect conclusions.
Not checking the mapping: Failing to check the mapping of the equation can lead to incorrect conclusions.
Not checking for repeated outputs: Failing to check for repeated outputs can lead to incorrect conclusions.

Q: How do I use functions in real-world applications?

A: Functions are used in a wide range of real-world applications, including:

Science: Functions are used to model real-world phenomena, such as population growth, chemical reactions, and physical systems.
Engineering: Functions are used to design and optimize systems, such as electronic circuits, mechanical systems, and computer algorithms.
Economics: Functions are used to model economic systems, such as supply and demand curves, and to make predictions about economic outcomes.

In conclusion, functions are a fundamental concept in mathematics, and understanding how to identify them is essential in various fields. By following the steps outlined in this Q&A guide, you can better understand functions and how to identify them.