Which Function Has A Range Of Y \textless 3 Y \ \textless \ 3 Y \textless 3 ?A. Y = 3 ( 2 ) X Y = 3(2)^x Y = 3 ( 2 ) X B. Y = 2 ( 3 ) X Y = 2(3)^x Y = 2 ( 3 ) X C. Y = − ( 2 ) X + 3 Y = -(2)^x + 3 Y = − ( 2 ) X + 3 D. Y = ( 2 ) X − 3 Y = (2)^x - 3 Y = ( 2 ) X − 3

When dealing with exponential functions, it's essential to understand the concept of range and how it relates to the function's behavior. The range of a function is the set of all possible output values it can produce for the given input values. In this article, we'll explore the range of four different exponential functions and determine which one has a range of $y < 3$.

What is an Exponential Function?

An exponential function is a mathematical function of the form $y = a^x$, where $a$ is a positive real number and $x$ is the variable. The base $a$ determines the rate at which the function grows or decays. If $a > 1$, the function grows exponentially, and if $0 < a < 1$, the function decays exponentially.

Range of Exponential Functions

The range of an exponential function depends on the base $a$. If $a > 1$, the range is all positive real numbers, i.e., $y > 0$. If $0 < a < 1$, the range is all positive real numbers, i.e., $y > 0$. If $a = 1$, the function is a constant function, and its range is a single value.

Analyzing the Given Functions

We are given four exponential functions:

A. $y = 3(2)^x$ B. $y = 2(3)^x$ C. $y = -(2)^x + 3$ D. $y = (2)^x - 3$

We need to determine which function has a range of $y < 3$.

Function A: $y = 3(2)^x$

The base of this function is $2$, which is greater than $1$. Therefore, the function grows exponentially. To determine the range, we can analyze the behavior of the function as $x$ approaches positive and negative infinity.

As $x$ approaches positive infinity, $y$ approaches infinity. As $x$ approaches negative infinity, $y$ approaches $0$.

Since the function grows exponentially, its range is all positive real numbers, i.e., $y > 0$. Therefore, function A does not have a range of $y < 3$.

Function B: $y = 2(3)^x$

The base of this function is $3$, which is greater than $1$. Therefore, the function grows exponentially. To determine the range, we can analyze the behavior of the function as $x$ approaches positive and negative infinity.

As $x$ approaches positive infinity, $y$ approaches infinity. As $x$ approaches negative infinity, $y$ approaches $0$.

Since the function grows exponentially, its range is all positive real numbers, i.e., $y > 0$. Therefore, function B does not have a range of $y < 3$.

Function C: $y = -(2)^x + 3$

The base of this function is $2$, which is greater than $1$. Therefore, the function grows exponentially. However, the function is shifted down by $3$ units, which means that its range is limited to $y < 3$.

As $x$ approaches positive infinity, $y$ approaches $-3$. As $x$ approaches negative infinity, $y$ approaches $3$.

Since the function is shifted down by $3$ units, its range is limited to $y < 3$. Therefore, function C has a range of $y < 3$.

Function D: $y = (2)^x - 3$

The base of this function is $2$, which is greater than $1$. Therefore, the function grows exponentially. However, the function is shifted down by $3$ units, which means that its range is limited to $y < 3$.

As $x$ approaches positive infinity, $y$ approaches $-3$. As $x$ approaches negative infinity, $y$ approaches $-3$.

Since the function is shifted down by $3$ units, its range is limited to $y < 3$. Therefore, function D has a range of $y < 3$.

Conclusion

In conclusion, we have analyzed four exponential functions and determined which one has a range of $y < 3$. The functions are:

A. $y = 3(2)^x$ B. $y = 2(3)^x$ C. $y = -(2)^x + 3$ D. $y = (2)^x - 3$

Only functions C and D have a range of $y < 3$. Therefore, the correct answer is:

• C. $y = -(2)^x + 3$
• D. $y = (2)^x - 3$

Final Answer

In the previous article, we explored the range of four different exponential functions and determined which ones have a range of $y < 3$. In this article, we'll answer some frequently asked questions about exponential functions and their ranges.

Q: What is an exponential function?

A: An exponential function is a mathematical function of the form $y = a^x$, where $a$ is a positive real number and $x$ is the variable. The base $a$ determines the rate at which the function grows or decays.

Q: What is the range of an exponential function?

A: The range of an exponential function depends on the base $a$. If $a > 1$, the range is all positive real numbers, i.e., $y > 0$. If $0 < a < 1$, the range is all positive real numbers, i.e., $y > 0$. If $a = 1$, the function is a constant function, and its range is a single value.

Q: How do I determine the range of an exponential function?

A: To determine the range of an exponential function, you can analyze the behavior of the function as $x$ approaches positive and negative infinity. If the function grows exponentially, its range is all positive real numbers. If the function decays exponentially, its range is all positive real numbers. If the function is a constant function, its range is a single value.

Q: What is the difference between a growing and a decaying exponential function?

A: A growing exponential function has a base $a > 1$, and its value increases as $x$ increases. A decaying exponential function has a base $0 < a < 1$, and its value decreases as $x$ increases.

Q: Can an exponential function have a range of $y < 0$?

A: No, an exponential function cannot have a range of $y < 0$. This is because the base $a$ is always positive, and the function will always grow or decay exponentially.

Q: Can an exponential function have a range of $y = 0$?

A: Yes, an exponential function can have a range of $y = 0$. This occurs when the base $a = 1$, and the function is a constant function.

Q: How do I shift an exponential function up or down?

A: To shift an exponential function up or down, you can add or subtract a constant value from the function. For example, if you have the function $y = a^x$, you can shift it up by $b$ units by adding $b$ to the function, resulting in $y = a^x + b$. You can shift it down by $b$ units by subtracting $b$ from the function, resulting in $y = a^x - b$.

Q: Can I have multiple exponential functions with the same range?

A: Yes, you can have multiple exponential functions with the same range. For example, the functions $y = a^x$ and $y = b^x$ can have the same range if $a$ and $b$ are both greater than $1$ or both less than $1$.

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

A: To determine the range of a composite function, you need to analyze the behavior of the inner function as $x$ approaches positive and negative infinity. Then, you can use the result to determine the range of the outer function.

Conclusion

In conclusion, we have answered some frequently asked questions about exponential functions and their ranges. We hope this article has provided you with a better understanding of exponential functions and their behavior. If you have any more questions, feel free to ask!

Final Answer

The final answer is that exponential functions can have a range of $y < 3$, but only if the function is shifted down by a constant value.