What Is The Range Of The Equation Y = − 2 ( 9 ) X − 4 + 7 Y = -2(9)^{x-4} + 7 Y = − 2 ( 9 ) X − 4 + 7 ?A. ( ∞ , 7 (\infty, 7 ( ∞ , 7 ]B. ( − ∞ , 7 (-\infty, 7 ( − ∞ , 7 ]C. ( − ∞ , 2 (-\infty, 2 ( − ∞ , 2 ]D. ( 4 , ∞ (4, \infty ( 4 , ∞ ]

Introduction

When dealing with functions, understanding the range of an equation is crucial in determining the possible output values. In this case, we are given the equation $y = -2(9)^{x-4} + 7$ 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 Equation

The given equation is an exponential function in the form of $y = ab^x + c$, where $a$, $b$, and $c$ are constants. In this case, $a = -2$, $b = 9$, and $c = 7$. The equation can be rewritten as $y = -2(9)^{x-4} + 7$. This equation represents a function that takes an input value $x$ and produces an output value $y$.

Analyzing the Exponential Term

The exponential term $-2(9)^{x-4}$ is the key to understanding the behavior of the function. The base of the exponential term is $9$, which is greater than $1$. This means that as $x$ increases, the value of $-2(9)^{x-4}$ will also increase. Conversely, as $x$ decreases, the value of $-2(9)^{x-4}$ will decrease.

Finding the Range

To find the range of the equation, we need to determine the possible output values of the function. Since the exponential term $-2(9)^{x-4}$ can take on any value between $-\infty$ and $\infty$, we need to consider the constant term $7$ that is added to the exponential term.

Case 1: When $x$ is very large

When $x$ is very large, the value of $-2(9)^{x-4}$ will also be very large. However, since we are adding $7$ to the exponential term, the value of $y$ will be very large minus $7$. This means that as $x$ becomes very large, the value of $y$ will approach $-\infty$.

Case 2: When $x$ is very small

When $x$ is very small, the value of $-2(9)^{x-4}$ will be very small. However, since we are adding $7$ to the exponential term, the value of $y$ will be very small plus $7$. This means that as $x$ becomes very small, the value of $y$ will approach $7$.

Conclusion

Based on the analysis of the equation, we can conclude that the range of the equation $y = -2(9)^{x-4} + 7$ is $(-\infty, 7]$. This means that the function can produce any output value between $-\infty$ and $7$, inclusive.

Final Answer

The final answer is $\boxed{(-\infty, 7]}$.

Discussion

The range of a function is an important concept in mathematics, and it is essential to understand how to find the range of an equation. In this case, we used the properties of exponential functions to determine the range of the equation $y = -2(9)^{x-4} + 7$. The range of the equation is $(-\infty, 7]$, which means that the function can produce any output value between $-\infty$ and $7$, inclusive.

Related Topics

• Exponential functions
• Range of a function
• Properties of exponential functions

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Range of a Function" by Khan Academy
• [3] "Properties of Exponential Functions" by Wolfram MathWorld
  

Introduction

In our previous article, we discussed the range of the equation $y = -2(9)^{x-4} + 7$. We analyzed the equation and determined that the range of the equation is $(-\infty, 7]$. In this article, we will answer some frequently asked questions related to the range of the equation.

Q: What is the range of the equation $y = -2(9)^{x-4} + 7$?

A: The range of the equation $y = -2(9)^{x-4} + 7$ is $(-\infty, 7]$. This means that the function can produce any output value between $-\infty$ and $7$, inclusive.

Q: Why is the range of the equation $(-\infty, 7]$?

A: The range of the equation is $(-\infty, 7]$ because the exponential term $-2(9)^{x-4}$ can take on any value between $-\infty$ and $\infty$, and the constant term $7$ is added to the exponential term. As $x$ becomes very large, the value of $y$ approaches $-\infty$, and as $x$ becomes very small, the value of $y$ approaches $7$.

Q: Can the function produce any output value between $7$ and $\infty$?

A: No, the function cannot produce any output value between $7$ and $\infty$. The range of the equation is $(-\infty, 7]$, which means that the function can only produce output values between $-\infty$ and $7$, inclusive.

Q: How does the range of the equation change if the constant term is changed?

A: If the constant term is changed, the range of the equation will also change. For example, if the constant term is changed to $5$, the range of the equation will be $(-\infty, 5]$. If the constant term is changed to $10$, the range of the equation will be $(-\infty, 10]$.

Q: Can the range of the equation be changed by changing the base of the exponential term?

A: Yes, the range of the equation can be changed by changing the base of the exponential term. For example, if the base of the exponential term is changed to $4$, the range of the equation will be different. However, the range of the equation will still be dependent on the constant term and the properties of the exponential function.

Q: How does the range of the equation relate to the domain of the equation?

A: The range of the equation is related to the domain of the equation. The domain of the equation is the set of all possible input values $x$ that can produce a valid output value $y$. The range of the equation is the set of all possible output values $y$ that can be produced by the function.

Conclusion

In this article, we answered some frequently asked questions related to the range of the equation $y = -2(9)^{x-4} + 7$. We discussed the properties of the exponential function and how the range of the equation is affected by the constant term and the base of the exponential term. We also related the range of the equation to the domain of the equation.

Final Answer

The final answer is $\boxed{(-\infty, 7]}$.

Discussion

The range of a function is an important concept in mathematics, and it is essential to understand how to find the range of an equation. In this article, we used the properties of exponential functions to determine the range of the equation $y = -2(9)^{x-4} + 7$. The range of the equation is $(-\infty, 7]$, which means that the function can produce any output value between $-\infty$ and $7$, inclusive.

Related Topics

• Exponential functions
• Range of a function
• Properties of exponential functions
• Domain of a function

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Range of a Function" by Khan Academy
• [3] "Properties of Exponential Functions" by Wolfram MathWorld
• [4] "Domain of a Function" by Math Is Fun