What is the Range of f(x) = 3x + 9? {y | y < 9} {y | y > 9} {y | y > 3} {y | y < 3}

In mathematics, understanding the range of a function is a key step in analyzing how that function behaves with respect to its output values. The range refers to all possible values that a function’s output (y-values) can take when different inputs (x-values) are applied. For a linear function like f(x) = 3x + 9, the range can extend infinitely in one or both directions, depending on the input.

In this article, we’ll explore the question: what is the range of f(x) = 3x + 9? {y | y < 9} {y | y > 9} {y | y > 3} {y | y < 3}. By breaking down the behavior of this linear function, we will discuss how the range is affected under different conditions such as y < 9, y > 9, y > 3, and y < 3. Whether you are new to understanding ranges or looking for deeper insights into this function, this article will provide a clear explanation and examples.

What is the Range of a Function?

In simple terms, the range of a function refers to the set of all possible output values (y-values) the function can produce. The function f(x) = 3x + 9 is a linear function, meaning it has a straight line when graphed. This implies that, for every value of x, there is a corresponding y-value, and these y-values span infinitely in both positive and negative directions.

For this specific function, the range is all real numbers. The equation f(x) = 3x + 9 describes a straight line with a slope of 3 and a y-intercept of 9. As x increases, f(x) also increases, and as x decreases, f(x) decreases, meaning the y-values cover all numbers from negative infinity to positive infinity.

The standard range for a linear function like this one is:

• Range of f(x) = 3x + 9: y ∈ (-∞, ∞).

However, if restrictions are applied to the output, the range can be limited. For example, {y | y < 9} restricts the range to values less than 9.

How Do You Determine the Range of f(x) = 3x + 9?

Understanding the Linear Function

A linear function like f(x) = 3x + 9 produces a straight line when graphed. This straight line extends infinitely, meaning it can take any value for y depending on the input value of x. The key to determining the range of this function is recognizing its unbounded nature.

Finding the Range with No Restrictions

The range of the function in its default form is all real numbers. This is because, as x increases, the output y increases as well, and as x decreases, the output y decreases. Therefore, the range is:

• Range of f(x) = 3x + 9: y ∈ (-∞, ∞).

Adding Restrictions

If restrictions are applied, such as {y | y < 9}, the range will be limited. To determine the restricted range, examine the inequality provided. For example:

• {y | y < 9} means that the range includes all y-values less than 9.
• {y | y > 9} restricts the range to values greater than 9.
• {y | y > 3} allows only y-values greater than 3.
• {y | y < 3} restricts the range to values less than 3.

Each condition affects the range differently, allowing us to define the output more specifically.

Key Insights into the Range of f(x) = 3x + 9

Let’s break down the key insights into determining the range for this function:

• Linear Nature: The function f(x) = 3x + 9 is linear, meaning its graph is a straight line with no breaks or restrictions. The y-values increase and decrease infinitely as x changes.
• Unrestricted Range: Without any restrictions, the range is all real numbers, represented as y ∈ (-∞, ∞).
• Restricted Ranges:
  • {y | y < 9}: The range includes all y-values less than 9.
  • {y | y > 9}: The range includes all y-values greater than 9.
  • {y | y > 3}: The range includes all y-values greater than 3.
  • {y | y < 3}: The range includes all y-values less than 3.

By applying these restrictions, we can refine our understanding of the output values for the function.

Common Questions About Function Range

1. What is the range of f(x) = 3x + 9?

The range of f(x) = 3x + 9 is all real numbers, or y ∈ (-∞, ∞), because the function is linear and has no restrictions.

2. What happens to the range when restrictions are applied?

When restrictions such as {y | y < 9} or {y | y > 9} are applied, the range is limited to values below or above the specified number.

3. Why is the range important in understanding a function?

The range helps us understand the possible values that a function can output, which is critical for analyzing its behavior over different inputs.

4. Can linear functions have restricted ranges?

Yes, while the default range of a linear function is all real numbers, adding conditions can restrict the range.

Graphical Representation of the Range of f(x) = 3x + 9

Visualizing the function f(x) = 3x + 9 can help clarify its range. When graphed, the function forms a straight line with a slope of 3 and a y-intercept of 9. As the x-values increase or decrease, the y-values follow, covering all possible outputs.

Without any restrictions, the graph extends infinitely. However, applying conditions like {y | y < 9} limits the range, causing the graph to only show values below 9.

Conclusion

In conclusion, the range of the linear function f(x) = 3x + 9 is all real numbers, represented by y ∈ (-∞, ∞). The function can produce any y-value depending on the input x, meaning the range is unrestricted unless specific conditions are applied. When restrictions such as {y | y < 9} or {y | y > 9} are introduced, the range is limited to the specified values. Understanding the range is critical for analyzing the output behavior of the function under different conditions.

FAQ’s

Q. What is the range of f(x) = 3x + 9?
A. The range is all real numbers: y ∈ (-∞, ∞).

Q. What does {y | y < 9} mean in terms of range?
A. This notation means the range includes all y-values less than 9.

Q. Can the range of a linear function be restricted?
A. Yes, by applying conditions like {y | y > 9} or {y | y < 3}, you can limit the range.

Q. Why is the range of a function important?
A. The range defines the possible output values, helping to understand the function’s behavior.

Q. What is the graphical representation of f(x) = 3x + 9?
A. The graph is a straight line with a slope of 3 and a y-intercept of 9, extending infinitely unless restricted.