Find The Range Of The Function Y = 3 4 X + 17 Y=\frac{3}{4} X+17 Y = 4 3 ​ X + 17 When The Domain Is {-12,-4,8}.Find The Range Of The Function G ( X ) = − 4 X + 24 G(x)=-4 X+24 G ( X ) = − 4 X + 24 When The Domain Is {-7,-2,1}.

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Introduction

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In mathematics, the range of a function is the set of all possible output values it can produce for the given input values. When dealing with linear functions, finding the range is crucial in understanding the behavior of the function. In this article, we will explore how to find the range of two linear functions, $y=\frac{3}{4} x+17$ and $g(x)=-4 x+24$, given their respective domains.

Understanding Linear Functions

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A linear function is a polynomial function of degree one, which can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope represents the rate of change of the function, while the y-intercept represents the point where the function intersects the y-axis.

Finding the Range of $y=\frac{3}{4} x+17$

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To find the range of the function $y=\frac{3}{4} x+17$, we need to determine the set of all possible output values it can produce for the given input values in the domain {-12,-4,8}. Since the function is linear, we can use the following steps to find the range:

1. Find the minimum and maximum values of the domain: The minimum value in the domain is -12, and the maximum value is 8.
2. Plug in the minimum and maximum values into the function: We will substitute the minimum and maximum values into the function to find the corresponding output values.

Minimum Value

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To find the output value corresponding to the minimum input value (-12), we plug -12 into the function:

$$y = \frac{3}{4}(-12) + 17$$

$$y = -9 + 17$$

$$y = 8$$

Maximum Value

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To find the output value corresponding to the maximum input value (8), we plug 8 into the function:

$$y = \frac{3}{4}(8) + 17$$

$$y = 6 + 17$$

$$y = 23$$

Conclusion

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Based on the calculations above, we can conclude that the range of the function $y=\frac{3}{4} x+17$ is the set of all values between 8 and 23, inclusive. This can be represented as $[8, 23]$.

Finding the Range of $g(x)=-4 x+24$

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To find the range of the function $g(x)=-4 x+24$, we need to determine the set of all possible output values it can produce for the given input values in the domain {-7,-2,1}. Since the function is linear, we can use the following steps to find the range:

1. Find the minimum and maximum values of the domain: The minimum value in the domain is -7, and the maximum value is 1.
2. Plug in the minimum and maximum values into the function: We will substitute the minimum and maximum values into the function to find the corresponding output values.

Minimum Value

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To find the output value corresponding to the minimum input value (-7), we plug -7 into the function:

$$g(x) = -4(-7) + 24$$

$$g(x) = 28 + 24$$

$$g(x) = 52$$

Maximum Value

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To find the output value corresponding to the maximum input value (1), we plug 1 into the function:

$$g(x) = -4(1) + 24$$

$$g(x) = -4 + 24$$

$$g(x) = 20$$

Conclusion

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Based on the calculations above, we can conclude that the range of the function $g(x)=-4 x+24$ is the set of all values between 20 and 52, inclusive. This can be represented as $[20, 52]$.

Conclusion

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In conclusion, finding the range of linear functions is a crucial step in understanding their behavior. By following the steps outlined in this article, we can determine the range of a linear function given its domain. The range of the function $y=\frac{3}{4} x+17$ is $[8, 23]$, and the range of the function $g(x)=-4 x+24$ is $[20, 52]$.

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Introduction

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In our previous article, we explored how to find the range of two linear functions, $y=\frac{3}{4} x+17$ and $g(x)=-4 x+24$, given their respective domains. In this article, we will answer some frequently asked questions about finding the range of linear functions.

Q&A

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Q: What is the range of a linear function?

A: The range of a linear function is the set of all possible output values it can produce for the given input values.

Q: How do I find the range of a linear function?

A: To find the range of a linear function, you need to determine the set of all possible output values it can produce for the given input values. You can do this by plugging in the minimum and maximum values of the domain into the function.

Q: What if the domain has only one value?

A: If the domain has only one value, then the range of the function is a single point. For example, if the domain is {5}, then the range of the function $y = 2x + 3$ is {13}.

Q: Can the range of a linear function be empty?

A: No, the range of a linear function cannot be empty. This is because a linear function is a polynomial function of degree one, and it always has at least one output value for any given input value.

Q: How do I determine if the range of a linear function is bounded or unbounded?

A: To determine if the range of a linear function is bounded or unbounded, you need to examine the slope of the function. If the slope is positive, then the range is unbounded. If the slope is negative, then the range is also unbounded. If the slope is zero, then the range is bounded.

Q: Can the range of a linear function be a single point?

A: Yes, the range of a linear function can be a single point. For example, if the domain is {5} and the function is $y = 2x + 3$, then the range of the function is {13}.

Q: How do I find the range of a linear function with a fractional slope?

A: To find the range of a linear function with a fractional slope, you can use the same steps as before. For example, if the function is $y = \frac{3}{4}x + 17$ and the domain is {-12, -4, 8}, then the range of the function is ${8, 23}$.

Q: Can the range of a linear function be a finite set of values?

A: Yes, the range of a linear function can be a finite set of values. For example, if the domain is {5, 10, 15} and the function is $y = 2x + 3$, then the range of the function is {13, 23, 33}.

Conclusion

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In conclusion, finding the range of linear functions is a crucial step in understanding their behavior. By following the steps outlined in this article, you can determine the range of a linear function given its domain. We hope this Q&A article has been helpful in answering some of the most frequently asked questions about finding the range of linear functions.

Additional Resources

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• Linear Functions
• Range of a Function
• Linear Algebra

Final Thoughts

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Finding the range of linear functions is an essential concept in mathematics, and it has many real-world applications. By understanding how to find the range of linear functions, you can solve a wide range of problems in mathematics, science, and engineering. We hope this article has been helpful in providing you with a deeper understanding of this concept.