What Are The Domain, Range, And Asymptote Of $h(x)=(1.4)^x+5$?A. Domain: $\{x \mid X \text{ Is A Real Number}\}$; Range: $\{y \mid Y \ \textgreater \ 5\}$; Asymptote: $y = 5$B. Domain: $\{x \mid X \

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Introduction

When dealing with functions, it's essential to understand the domain, range, and asymptotes. The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values. Asymptotes, on the other hand, are lines or curves that the function approaches as the input values get arbitrarily large or small. In this article, we will explore the domain, range, and asymptote of the function $h(x)=(1.4)^x+5$.

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 function $h(x)=(1.4)^x+5$, the base of the exponential term is 1.4, which is a positive real number. Since the exponent can be any real number, the domain of the function is the set of all real numbers. Therefore, the domain of the function is {x∣x is a real number}\{x \mid x \text{ is a real number}\}.

Range of the Function

The range of a function is the set of all possible output values. To determine the range of the function $h(x)=(1.4)^x+5$, we need to consider the behavior of the exponential term as the input values get arbitrarily large or small. As the input values get arbitrarily large, the exponential term (1.4)x(1.4)^x gets arbitrarily large, and when added to 5, the output values get arbitrarily large. Similarly, as the input values get arbitrarily small, the exponential term (1.4)x(1.4)^x gets arbitrarily close to 0, and when added to 5, the output values get arbitrarily close to 5. Therefore, the range of the function is the set of all real numbers greater than 5, which can be represented as {y∣y \textgreater 5}\{y \mid y \ \textgreater \ 5\}.

Asymptote of the Function

An asymptote is a line or curve that the function approaches as the input values get arbitrarily large or small. In the case of the function $h(x)=(1.4)^x+5$, the exponential term (1.4)x(1.4)^x gets arbitrarily close to 0 as the input values get arbitrarily small. When added to 5, the output values get arbitrarily close to 5. Therefore, the function has a horizontal asymptote at y=5y = 5.

Conclusion

In conclusion, the domain of the function $h(x)=(1.4)^x+5$ is the set of all real numbers, the range is the set of all real numbers greater than 5, and the function has a horizontal asymptote at y=5y = 5. Therefore, the correct answer is:

A. Domain: {x∣x is a real number}\{x \mid x \text{ is a real number}\}; Range: {y∣y \textgreater 5}\{y \mid y \ \textgreater \ 5\}; Asymptote: y=5y = 5

Final Answer

The final answer is A.

Introduction

In our previous article, we explored the domain, range, and asymptote of the function $h(x)=(1.4)^x+5$. In this article, we will answer some frequently asked questions related to the domain, range, and asymptote of this function.

Q: What is the domain of the function $h(x)=(1.4)^x+5$?

A: The domain of the function $h(x)=(1.4)^x+5$ is the set of all real numbers. This is because the base of the exponential term is 1.4, which is a positive real number, and the exponent can be any real number.

Q: Why is the range of the function $h(x)=(1.4)^x+5$ all real numbers greater than 5?

A: The range of the function $h(x)=(1.4)^x+5$ is all real numbers greater than 5 because as the input values get arbitrarily large, the exponential term (1.4)x(1.4)^x gets arbitrarily large, and when added to 5, the output values get arbitrarily large. Similarly, as the input values get arbitrarily small, the exponential term (1.4)x(1.4)^x gets arbitrarily close to 0, and when added to 5, the output values get arbitrarily close to 5.

Q: What is the horizontal asymptote of the function $h(x)=(1.4)^x+5$?

A: The horizontal asymptote of the function $h(x)=(1.4)^x+5$ is y=5y = 5. This is because as the input values get arbitrarily small, the exponential term (1.4)x(1.4)^x gets arbitrarily close to 0, and when added to 5, the output values get arbitrarily close to 5.

Q: Can the function $h(x)=(1.4)^x+5$ take on any value greater than 5?

A: Yes, the function $h(x)=(1.4)^x+5$ can take on any value greater than 5. This is because as the input values get arbitrarily large, the exponential term (1.4)x(1.4)^x gets arbitrarily large, and when added to 5, the output values get arbitrarily large.

Q: Can the function $h(x)=(1.4)^x+5$ take on any value less than 5?

A: No, the function $h(x)=(1.4)^x+5$ cannot take on any value less than 5. This is because as the input values get arbitrarily small, the exponential term (1.4)x(1.4)^x gets arbitrarily close to 0, and when added to 5, the output values get arbitrarily close to 5.

Q: Is the function $h(x)=(1.4)^x+5$ a one-to-one function?

A: No, the function $h(x)=(1.4)^x+5$ is not a one-to-one function. This is because the function is not strictly increasing or decreasing, and it takes on the same value for different input values.

Q: Can the function $h(x)=(1.4)^x+5$ be inverted?

A: No, the function $h(x)=(1.4)^x+5$ cannot be inverted. This is because the function is not one-to-one, and it takes on the same value for different input values.

Conclusion

In conclusion, we have answered some frequently asked questions related to the domain, range, and asymptote of the function $h(x)=(1.4)^x+5$. We hope that this article has provided a better understanding of the properties of this function.

Final Answer

The final answer is that the domain of the function $h(x)=(1.4)^x+5$ is the set of all real numbers, the range is the set of all real numbers greater than 5, and the function has a horizontal asymptote at y=5y = 5.