Choose The Function That Has Domain X ≠ − 3 X \neq -3 X  = − 3 And Range Y ≠ 2 Y \neq 2 Y  = 2 .A. F ( X ) = X + 2 X + 3 F(x)=\frac{x+2}{x+3} F ( X ) = X + 3 X + 2 ​ B. F ( X ) = 2 X + 1 X + 3 F(x)=\frac{2x+1}{x+3} F ( X ) = X + 3 2 X + 1 ​ C. F ( X ) = X − 3 X + 2 F(x)=\frac{x-3}{x+2} F ( X ) = X + 2 X − 3 ​

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When dealing with functions, it's essential to understand the concept of domain and range. The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce. In this article, we will explore the domain and range of three given functions and determine which one has a domain of x≠−3x \neq -3 and a range of y≠2y \neq 2.

Understanding the Domain and Range of a Function

The domain of a function is the set of all possible input values that the function can accept. This means that the function is defined for all values of x in the domain. On the other hand, the range of a function is the set of all possible output values that the function can produce. This means that the function can produce all values of y in the range.

Analyzing the Given Functions

We are given three functions:

A. f(x)=x+2x+3f(x)=\frac{x+2}{x+3} B. f(x)=2x+1x+3f(x)=\frac{2x+1}{x+3} C. f(x)=x−3x+2f(x)=\frac{x-3}{x+2}

We need to determine which function has a domain of x≠−3x \neq -3 and a range of y≠2y \neq 2.

Function A: f(x)=x+2x+3f(x)=\frac{x+2}{x+3}

To determine the domain of function A, we need to find the values of x that make the denominator equal to zero. The denominator is x+3x+3, so we set it equal to zero and solve for x:

x+3=0x+3=0

x=−3x=-3

This means that the function is not defined at x=−3x=-3, so the domain of function A is x≠−3x \neq -3.

To determine the range of function A, we need to find the values of y that the function can produce. We can do this by finding the inverse of the function. The inverse of function A is:

y=3y+2y+1y=\frac{3y+2}{y+1}

Solving for y, we get:

y=3y+2y+1y=\frac{3y+2}{y+1}

y(y+1)=3y+2y(y+1)=3y+2

y2+y=3y+2y^2+y=3y+2

y2−2y−2=0y^2-2y-2=0

Solving for y, we get:

y=2±4+82y=\frac{2\pm\sqrt{4+8}}{2}

y=2±122y=\frac{2\pm\sqrt{12}}{2}

y=2±232y=\frac{2\pm2\sqrt{3}}{2}

y=1±3y=1\pm\sqrt{3}

This means that the range of function A is y=1±3y=1\pm\sqrt{3}, which does not include y=2y=2.

Function B: f(x)=2x+1x+3f(x)=\frac{2x+1}{x+3}

To determine the domain of function B, we need to find the values of x that make the denominator equal to zero. The denominator is x+3x+3, so we set it equal to zero and solve for x:

x+3=0x+3=0

x=−3x=-3

This means that the function is not defined at x=−3x=-3, so the domain of function B is x≠−3x \neq -3.

To determine the range of function B, we need to find the values of y that the function can produce. We can do this by finding the inverse of the function. The inverse of function B is:

y=3y−12yy=\frac{3y-1}{2y}

Solving for y, we get:

y=3y−12yy=\frac{3y-1}{2y}

2y2=3y−12y^2=3y-1

2y2−3y+1=02y^2-3y+1=0

Solving for y, we get:

y=3±9−84y=\frac{3\pm\sqrt{9-8}}{4}

y=3±14y=\frac{3\pm1}{4}

y=3±14y=\frac{3\pm1}{4}

y=12y=\frac{1}{2} or y=12y=\frac{1}{2}

This means that the range of function B is y=12y=\frac{1}{2}, which does not include y=2y=2.

Function C: f(x)=x−3x+2f(x)=\frac{x-3}{x+2}

To determine the domain of function C, we need to find the values of x that make the denominator equal to zero. The denominator is x+2x+2, so we set it equal to zero and solve for x:

x+2=0x+2=0

x=−2x=-2

This means that the function is not defined at x=−2x=-2, so the domain of function C is x≠−2x \neq -2.

To determine the range of function C, we need to find the values of y that the function can produce. We can do this by finding the inverse of the function. The inverse of function C is:

y=2y+3y−1y=\frac{2y+3}{y-1}

Solving for y, we get:

y=2y+3y−1y=\frac{2y+3}{y-1}

y(y−1)=2y+3y(y-1)=2y+3

y2−y=2y+3y^2-y=2y+3

y2−3y−3=0y^2-3y-3=0

Solving for y, we get:

y=3±9+122y=\frac{3\pm\sqrt{9+12}}{2}

y=3±212y=\frac{3\pm\sqrt{21}}{2}

y=3±212y=\frac{3\pm\sqrt{21}}{2}

This means that the range of function C is y=3±212y=\frac{3\pm\sqrt{21}}{2}, which does not include y=2y=2.

Conclusion

Based on our analysis, we can conclude that function A has a domain of x≠−3x \neq -3 and a range of y≠2y \neq 2. Therefore, the correct answer is:

A. f(x)=x+2x+3f(x)=\frac{x+2}{x+3}

This function has a domain of x≠−3x \neq -3 and a range of y≠2y \neq 2, making it the correct choice.

References

In our previous article, we explored the domain and range of three given functions and determined which one has a domain of x≠−3x \neq -3 and a range of y≠2y \neq 2. In this article, we will answer some frequently asked questions about the domain and range of functions.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) that the function can accept. This means that the function is defined for all values of x in the domain.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values (y-values) that the function can produce. This means that the function can produce all values of y in the range.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to find the values of x that make the denominator equal to zero. The denominator is the part of the function that is in the denominator, and it cannot be equal to zero.

Q: How do I determine the range of a function?

A: To determine the range of a function, you need to find the values of y that the function can produce. You can do this by finding the inverse of the function.

Q: What is the inverse of a function?

A: The inverse of a function is a function that undoes the action of the original function. In other words, if the original function takes an input x and produces an output y, then the inverse function takes the output y and produces the input x.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you need to swap the x and y variables and then solve for y. This will give you the inverse function.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range of a function is the set of all possible output values (y-values) that the function can produce.

Q: Can a function have a domain of all real numbers?

A: Yes, a function can have a domain of all real numbers. This means that the function is defined for all values of x, and there are no restrictions on the input values.

Q: Can a function have a range of all real numbers?

A: Yes, a function can have a range of all real numbers. This means that the function can produce all values of y, and there are no restrictions on the output values.

Q: What is the relationship between the domain and range of a function?

A: The domain and range of a function are related in that the domain of a function determines the range of the function. In other words, the range of a function is determined by the values of x that are in the domain of the function.

Q: Can a function have a domain and range that are the same?

A: Yes, a function can have a domain and range that are the same. This means that the function is defined for all values of x, and the function can produce all values of y.

Conclusion

In this article, we have answered some frequently asked questions about the domain and range of functions. We have also discussed how to determine the domain and range of a function, and how to find the inverse of a function. We hope that this article has been helpful in understanding the domain and range of functions.

References