What Is The Range Of F ( X ) = ( 3 4 ) X − 4 F(x)=\left(\frac{3}{4}\right)^x-4 F ( X ) = ( 4 3 ​ ) X − 4 ?A. { Y ∣ Y \textgreater − 4 } \{y \mid Y \ \textgreater \ -4\} { Y ∣ Y \textgreater − 4 } B. { Y ∣ Y \textgreater 3 4 } \left\{y \left\lvert\, Y \ \textgreater \ \frac{3}{4}\right.\right\} { Y ​ Y \textgreater 4 3 ​ } C. { Y ∣ Y \textless − 4 } \{y \mid Y \ \textless \ -4\} { Y ∣ Y \textless − 4 } D.

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Introduction

When dealing with functions, understanding the range is crucial in determining the possible output values. In this case, we are given the function f(x)=(34)x−4f(x)=\left(\frac{3}{4}\right)^x-4 and asked to find its range. The range of a function is the set of all possible output values it can produce for the given input values. In other words, it is the set of all possible y-values that the function can take.

Understanding the Function

To find the range of the function f(x)=(34)x−4f(x)=\left(\frac{3}{4}\right)^x-4, we need to understand the behavior of the function as x varies. The function is an exponential function with a base of 34\frac{3}{4}, and it is shifted down by 4 units. This means that the function will always be less than or equal to -4, and it will approach -4 as x approaches infinity.

Finding the Range

To find the range of the function, we need to determine the set of all possible output values it can produce. Since the function is an exponential function with a base of 34\frac{3}{4}, it will always be positive for positive values of x. However, since the function is shifted down by 4 units, it will always be less than or equal to -4.

Analyzing the Options

Now that we have a good understanding of the function, let's analyze the options given:

A. {y∣y \textgreater −4}\{y \mid y \ \textgreater \ -4\}

This option suggests that the range of the function is all values greater than -4. However, we know that the function will always be less than or equal to -4, so this option is not correct.

B. {y∣ y \textgreater 34}\left\{y \left\lvert\, y \ \textgreater \ \frac{3}{4}\right.\right\}

This option suggests that the range of the function is all values greater than 34\frac{3}{4}. However, we know that the function will always be less than or equal to -4, so this option is not correct.

C. {y∣y \textless −4}\{y \mid y \ \textless \ -4\}

This option suggests that the range of the function is all values less than -4. However, we know that the function will always be less than or equal to -4, so this option is not correct.

D. {y∣y \textless −4}\boxed{\{y \mid y \ \textless \ -4\}}

This option suggests that the range of the function is all values less than -4. This is the correct answer, as the function will always be less than or equal to -4.

Conclusion

In conclusion, the range of the function f(x)=(34)x−4f(x)=\left(\frac{3}{4}\right)^x-4 is all values less than -4. This is because the function is an exponential function with a base of 34\frac{3}{4}, and it is shifted down by 4 units. Therefore, the correct answer is option D.

Final Answer

The final answer is {y∣y \textless −4}\boxed{\{y \mid y \ \textless \ -4\}}.

Introduction

In our previous article, we discussed the range of the function f(x)=(34)x−4f(x)=\left(\frac{3}{4}\right)^x-4. We concluded that the range of the function is all values less than -4. However, we received many questions from readers who were unsure about the concept of range and how to apply it to different types of functions. In this article, we will answer some of the most frequently asked questions about the range of the function f(x)=(34)x−4f(x)=\left(\frac{3}{4}\right)^x-4.

Q&A

Q: What is the range of a function?

A: The range of a function is the set of all possible output values it can produce for the given input values. In other words, it is the set of all possible y-values that the function can take.

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

A: To find the range of a function, you need to determine the set of all possible output values it can produce. This can be done by analyzing the function's behavior as x varies. For example, if the function is an exponential function with a base of 34\frac{3}{4}, it will always be positive for positive values of x.

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

A: The domain of a function is the set of all possible input values it can take, while the range of a function is the set of all possible output values it can produce. In other words, the domain is the set of all possible x-values, while the range is the set of all possible y-values.

Q: Can the range of a function be a single value?

A: Yes, the range of a function can be a single value. For example, if the function is a constant function, its range will be a single value.

Q: Can the range of a function be an interval?

A: Yes, the range of a function can be an interval. For example, if the function is a linear function, its range will be an interval.

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

A: To determine if the range of a function is bounded or unbounded, you need to analyze the function's behavior as x varies. If the function approaches a finite value as x approaches infinity, the range is bounded. If the function approaches infinity as x approaches infinity, the range is unbounded.

Q: Can the range of a function be a union of intervals?

A: Yes, the range of a function can be a union of intervals. For example, if the function is a piecewise function, its range will be a union of intervals.

Conclusion

In conclusion, the range of a function is the set of all possible output values it can produce for the given input values. To find the range of a function, you need to determine the set of all possible output values it can produce by analyzing the function's behavior as x varies. The range of a function can be a single value, an interval, or a union of intervals. We hope this article has helped you understand the concept of range and how to apply it to different types of functions.

Final Answer

The final answer is {y∣y \textless −4}\boxed{\{y \mid y \ \textless \ -4\}}.