Which Are True Of The Function $f(x)=49\left(\frac{1}{7}\right)^x$? Select Three Options.A. The Domain Is The Set Of All Real Numbers.B. The Range Is The Set Of All Real Numbers.C. The Domain Is $x\ \textgreater \ 0$.D. The Range Is

The given function is $f(x)=49\left(\frac{1}{7}\right)^x$. This function represents an exponential relationship between the input variable $x$ and the output value $f(x)$. To determine the properties of this function, we need to analyze its behavior and characteristics.

Domain of the Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of the given function, we need to consider the restrictions on the input variable $x$.

• Option A: The domain is the set of all real numbers. This option suggests that the function is defined for all real numbers, including both positive and negative values. However, we need to examine the function more closely to determine its domain.
• Option C: The domain is $x\ \textgreater \ 0$. This option restricts the domain to only positive values of $x$. We need to evaluate whether this restriction is valid.

To determine the domain, let's consider the behavior of the function as $x$ approaches negative infinity. As $x$ becomes increasingly negative, the value of $\left(\frac{1}{7}\right)^x$ approaches infinity. This is because the base $\frac{1}{7}$ is less than 1, and raising a number less than 1 to a negative power results in a value greater than 1.

However, the function is not defined for $x=0$ because the expression $\left(\frac{1}{7}\right)^0$ is equal to 1, and multiplying 1 by 49 results in 49, not 0. Therefore, the domain of the function is not the set of all real numbers, and it is not restricted to only positive values of $x$.

The correct statement about the domain is that it is the set of all real numbers except 0. However, this option is not available among the choices provided.

Range of the Function

The range of a function is the set of all possible output values for which the function is defined. In the case of the given function, we need to consider the behavior of the function as $x$ approaches positive and negative infinity.

• Option B: The range is the set of all real numbers. This option suggests that the function is defined for all real numbers, including both positive and negative values. However, we need to examine the function more closely to determine its range.
• Option D: The range is... This option is incomplete, and we cannot evaluate it without more information.

To determine the range, let's consider the behavior of the function as $x$ approaches positive and negative infinity. As $x$ becomes increasingly positive, the value of $\left(\frac{1}{7}\right)^x$ approaches 0. This is because the base $\frac{1}{7}$ is less than 1, and raising a number less than 1 to a positive power results in a value less than 1.

As $x$ becomes increasingly negative, the value of $\left(\frac{1}{7}\right)^x$ approaches infinity. This is because the base $\frac{1}{7}$ is less than 1, and raising a number less than 1 to a negative power results in a value greater than 1.

However, the function is not defined for $x=0$ because the expression $\left(\frac{1}{7}\right)^0$ is equal to 1, and multiplying 1 by 49 results in 49, not 0. Therefore, the range of the function is not the set of all real numbers.

The correct statement about the range is that it is the set of all real numbers except 0. However, this option is not available among the choices provided.

Conclusion

In conclusion, the domain of the function $f(x)=49\left(\frac{1}{7}\right)^x$ is the set of all real numbers except 0, and the range is the set of all real numbers except 0. However, these options are not available among the choices provided.

The correct answer is not among the options provided. However, based on the analysis above, we can conclude that the domain is not the set of all real numbers, and the range is not the set of all real numbers.

Final Answer

Q: What is the domain of the function $f(x)=49\left(\frac{1}{7}\right)^x$?

A: The domain of the function is the set of all real numbers except 0. This is because the function is not defined for $x=0$ because the expression $\left(\frac{1}{7}\right)^0$ is equal to 1, and multiplying 1 by 49 results in 49, not 0.

Q: Why is the domain not the set of all real numbers?

A: The domain is not the set of all real numbers because the function is not defined for $x=0$. As $x$ approaches negative infinity, the value of $\left(\frac{1}{7}\right)^x$ approaches infinity, and the function is not defined for this value.

Q: What is the range of the function $f(x)=49\left(\frac{1}{7}\right)^x$?

A: The range of the function is the set of all real numbers except 0. This is because the function approaches 0 as $x$ approaches positive infinity, and the function approaches infinity as $x$ approaches negative infinity.

Q: Why is the range not the set of all real numbers?

A: The range is not the set of all real numbers because the function is not defined for $x=0$, and the function approaches 0 as $x$ approaches positive infinity.

Q: What happens to the function as $x$ approaches positive infinity?

A: As $x$ approaches positive infinity, the value of $\left(\frac{1}{7}\right)^x$ approaches 0. This is because the base $\frac{1}{7}$ is less than 1, and raising a number less than 1 to a positive power results in a value less than 1.

Q: What happens to the function as $x$ approaches negative infinity?

A: As $x$ approaches negative infinity, the value of $\left(\frac{1}{7}\right)^x$ approaches infinity. This is because the base $\frac{1}{7}$ is less than 1, and raising a number less than 1 to a negative power results in a value greater than 1.

Q: How does the function behave for positive values of $x$?

A: For positive values of $x$, the function approaches 0 as $x$ increases. This is because the base $\frac{1}{7}$ is less than 1, and raising a number less than 1 to a positive power results in a value less than 1.

Q: How does the function behave for negative values of $x$?

A: For negative values of $x$, the function approaches infinity as $x$ decreases. This is because the base $\frac{1}{7}$ is less than 1, and raising a number less than 1 to a negative power results in a value greater than 1.

Q: What is the behavior of the function at $x=0$?

A: The function is not defined at $x=0$ because the expression $\left(\frac{1}{7}\right)^0$ is equal to 1, and multiplying 1 by 49 results in 49, not 0.

Q: What is the behavior of the function as $x$ approaches infinity?

A: As $x$ approaches infinity, the value of $\left(\frac{1}{7}\right)^x$ approaches 0. This is because the base $\frac{1}{7}$ is less than 1, and raising a number less than 1 to a positive power results in a value less than 1.

Q: What is the behavior of the function as $x$ approaches negative infinity?

A: As $x$ approaches negative infinity, the value of $\left(\frac{1}{7}\right)^x$ approaches infinity. This is because the base $\frac{1}{7}$ is less than 1, and raising a number less than 1 to a negative power results in a value greater than 1.

Q: What is the significance of the base $\frac{1}{7}$ in the function?

A: The base $\frac{1}{7}$ is significant because it determines the behavior of the function. Since the base is less than 1, the function approaches 0 as $x$ approaches positive infinity, and the function approaches infinity as $x$ approaches negative infinity.

Q: What is the significance of the exponent $x$ in the function?

A: The exponent $x$ is significant because it determines the rate at which the function approaches 0 or infinity. As $x$ increases, the function approaches 0 more quickly, and as $x$ decreases, the function approaches infinity more quickly.

Q: What is the relationship between the function and the number 49?

A: The number 49 is a constant factor in the function, and it determines the scale of the function. The function can be rewritten as $f(x)=49\left(\frac{1}{7}\right)^x=7^2\left(\frac{1}{7}\right)^x$, which shows that the number 49 is equal to $7^2$.