Which Of The Following Best Describes The Range Of The Function F ( X ) = 2 X − 3 F(x)=2^x-3 F ( X ) = 2 X − 3 ?a. Y ≥ 0 Y \geq 0 Y ≥ 0 B. Y \textgreater − 3 Y \ \textgreater \ -3 Y \textgreater − 3 C. Y \textless − 3 Y \ \textless \ -3 Y \textless − 3 D. Y ≤ − 3 Y \leq -3 Y ≤ − 3

Mar 13, 2025 by ADMIN 286 views

Understanding the Range of a Function: A Case Study with $f(x)=2^x-3$

When dealing with functions, understanding their range is crucial in various mathematical and real-world applications. 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 will explore the range of the function $f(x)=2^x-3$ and determine which of the given options best describes it.

What is the Range of a Function?

The range of a function is the set of all possible output values it can produce for the given input values. It is a subset of the codomain of the function, which is the set of all possible output values. The range of a function can be determined by analyzing its graph, finding the minimum and maximum values, or using mathematical techniques such as calculus.

Analyzing the Function $f(x)=2^x-3$

The given function is $f(x)=2^x-3$ . To determine its range, we need to analyze its behavior as $x$ varies. Since the function involves an exponential term, we can expect it to grow rapidly as $x$ increases.

Finding the Minimum Value

To find the minimum value of the function, we can take the derivative of $f(x)$ with respect to $x$ and set it equal to zero. The derivative of $f(x)$ is given by:

f'(x) = 2^x \ln 2

Setting $f'(x) = 0$ , we get:

2^x \ln 2 = 0

Since $2^x$ is always positive, the only way for the derivative to be zero is if $\ln 2 = 0$ . However, $\ln 2$ is a constant and cannot be zero. Therefore, the function has no critical points and no minimum value.

Finding the Maximum Value

Since the function has no critical points, we can conclude that it has no maximum value either. The function will continue to grow as $x$ increases, and there is no upper bound to its values.

Determining the Range

Based on the analysis above, we can conclude that the range of the function $f(x)=2^x-3$ is all real numbers greater than or equal to $-3$ . This is because the function will continue to grow as $x$ increases, and there is no upper bound to its values. The minimum value of the function is $-3$ , which occurs when $x = -\infty$ .

In conclusion, the range of the function $f(x)=2^x-3$ is all real numbers greater than or equal to $-3$ . This is because the function will continue to grow as $x$ increases, and there is no upper bound to its values. The minimum value of the function is $-3$ , which occurs when $x = -\infty$ .

Based on the analysis above, the correct answer is:

a. $y \geq 0$ : This is incorrect because the range of the function is not all non-negative values.
b. $y \ \textgreater \ -3$ : This is correct because the range of the function is all real numbers greater than or equal to $-3$ .
c. $y \ \textless \ -3$ : This is incorrect because the range of the function is not all negative values less than $-3$ .
d. $y \leq -3$ : This is incorrect because the range of the function is not all values less than or equal to $-3$ .

Therefore, the correct answer is b. $y \ \textgreater \ -3$ .
Frequently Asked Questions: Understanding the Range of a Function

In our previous article, we explored the range of the function $f(x)=2^x-3$ and determined that it is all real numbers greater than or equal to $-3$ . In this article, we will answer some frequently asked questions related to the range of a function.

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, while the range of a function is the set of all possible output values. 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: How do I determine the range of a function?

A: There are several ways to determine the range of a function, including:

Analyzing the graph of the function
Finding the minimum and maximum values of the function
Using mathematical techniques such as calculus
Using the definition of the function to determine the range

Q: What is the range of a linear function?

A: The range of a linear function is all real numbers. This is because a linear function is a straight line, and a straight line can take on any value.

Q: What is the range of an exponential function?

A: The range of an exponential function is all positive real numbers. This is because an exponential function grows rapidly as the input value increases, and there is no upper bound to its values.

Q: What is the range of a quadratic function?

A: The range of a quadratic function is all real numbers. This is because a quadratic function is a parabola, and a parabola can take on any value.

Q: How do I determine the range of a function with a square root?

A: To determine the range of a function with a square root, you need to consider the domain of the function. The square root function is only defined for non-negative values, so the range of a function with a square root will be all non-negative real numbers.

Q: What is the range of a function with a logarithm?

A: The range of a function with a logarithm is all real numbers. This is because a logarithmic function can take on any value, and there is no upper bound to its values.

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

A: To determine the range of a function with a trigonometric function, you need to consider the properties of the trigonometric function. For example, the sine function has a range of $[-1, 1]$ , while the cosine function has a range of $[-1, 1]$ .

In conclusion, the range of a function is an important concept in mathematics that can be used to determine the possible output values of a function. By understanding the range of a function, you can better analyze and solve problems involving functions.

Assuming that the range of a function is all real numbers
Failing to consider the domain of a function
Not analyzing the graph of a function
Not using mathematical techniques such as calculus to determine the range of a function

Use the definition of a function to determine its range
Analyze the graph of a function to determine its range
Use mathematical techniques such as calculus to determine the range of a function
Consider the domain of a function when determining its range