What Are The Domain And Range Of The Function F ( X ) = − Log ⁡ ( 5 − X ) + 9 F(x) = -\log (5-x) + 9 F ( X ) = − Lo G ( 5 − X ) + 9 ?A. Domain: X \textless 5 X \ \textless \ 5 X \textless 5 , Range: Y ≥ 9 Y \geq 9 Y ≥ 9 B. Domain: X \textless 5 X \ \textless \ 5 X \textless 5 , Range: ( − ∞ , ∞ (-\infty, \infty ( − ∞ , ∞ ] C. Domain: $x

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Introduction

In mathematics, functions are used to describe the relationship between variables. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In this article, we will explore the domain and range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9. This function involves a logarithmic term, which requires careful consideration to determine its domain and range.

The Logarithmic Function

The logarithmic function is defined as:

log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = x

where bb is the base of the logarithm, and xx is the input value. The logarithmic function is used to describe exponential relationships between variables.

The Domain of the Function

To determine the domain of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9, we need to consider the restrictions on the input value xx. The logarithmic term log⁡(5−x)\log (5-x) requires that the input value 5−x5-x must be greater than zero, since the logarithm of a non-positive number is undefined.

5−x>0  ⟺  x<55-x > 0 \iff x < 5

Therefore, the domain of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 is x<5x < 5.

The Range of the Function

To determine the range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9, we need to consider the possible output values. The logarithmic term log⁡(5−x)\log (5-x) can take on any real value, since the logarithm of a positive number can be any real number. However, the negative sign in front of the logarithmic term means that the output value will be negative.

f(x)=−log⁡(5−x)+9≤9f(x) = -\log (5-x) + 9 \leq 9

Therefore, the range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 is y≤9y \leq 9.

Conclusion

In conclusion, the domain of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 is x<5x < 5, and the range is y≤9y \leq 9. This means that the function is defined for all input values less than 5, and the output values can range from negative infinity to 9.

Answer

The correct answer is:

A. Domain: x<5x < 5, Range: y≤9y \leq 9

This answer is consistent with the analysis presented above.

Discussion

The domain and range of a function are critical components of its definition. Understanding the domain and range of a function is essential for working with it, as it determines the possible input and output values. In this article, we have explored the domain and range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9, and have shown that the domain is x<5x < 5 and the range is y≤9y \leq 9. This analysis is important for anyone working with logarithmic functions, as it provides a clear understanding of the possible input and output values.

Additional Resources

For further information on the domain and range of functions, see the following resources:

References

License

Introduction

In our previous article, we explored the domain and range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9. In this article, we will answer some common questions related to the domain and range of this function.

Q: What is the domain of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9?

A: The domain of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 is x<5x < 5. This means that the function is defined for all input values less than 5.

Q: Why is the domain x<5x < 5?

A: The domain x<5x < 5 is due to the restriction on the input value 5−x5-x. Since the logarithmic term log⁡(5−x)\log (5-x) requires that the input value 5−x5-x must be greater than zero, we have 5−x>0  ⟺  x<55-x > 0 \iff x < 5.

Q: What is the range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9?

A: The range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 is y≤9y \leq 9. This means that the output values can range from negative infinity to 9.

Q: Why is the range y≤9y \leq 9?

A: The range y≤9y \leq 9 is due to the negative sign in front of the logarithmic term. Since the logarithmic term log⁡(5−x)\log (5-x) can take on any real value, the negative sign means that the output value will be negative. Additionally, the constant term +9+9 means that the output value will be less than or equal to 9.

Q: Can the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 take on any value in the range y≤9y \leq 9?

A: No, the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 cannot take on any value in the range y≤9y \leq 9. Since the function is defined for all input values less than 5, the output values will be limited to the range y≤9y \leq 9.

Q: How does the domain and range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 relate to the graph of the function?

A: The domain and range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 are reflected in the graph of the function. The graph of the function will be a curve that is defined for all input values less than 5, and the output values will range from negative infinity to 9.

Q: Can the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 be used to model real-world phenomena?

A: Yes, the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 can be used to model real-world phenomena. For example, the function could be used to model the relationship between the temperature of a substance and the time it takes for the substance to cool down.

Conclusion

In conclusion, the domain and range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9 are critical components of its definition. Understanding the domain and range of a function is essential for working with it, as it determines the possible input and output values. We hope that this Q&A article has provided a clear understanding of the domain and range of the function f(x)=−log⁡(5−x)+9f(x) = -\log (5-x) + 9.

Additional Resources

For further information on the domain and range of functions, see the following resources:

References

License

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