Understanding the Function and Its Properties

Oct 13, 2025 by miodeya.com 126 views

Let's Explore the Properties of the Function f(x) = (x-7)^2

Understanding the Function and Its Properties

The given function is f(x) = (x-7)^2. This is a quadratic function that represents a parabola with its vertex at (7, 0). The function is in the form of a perfect square, which means it can be rewritten as f(x) = (x-7)(x-7).

1. Largest Domain on Which f is One-to-One and Non-Decreasing

To find the largest domain on which f is one-to-one and non-decreasing, we need to consider the properties of the function.

One-to-One Function: A function is one-to-one if it passes the horizontal line test, meaning that no horizontal line intersects the graph of the function in more than one place. In other words, each value of the function corresponds to exactly one value of the input.
Non-Decreasing Function: A function is non-decreasing if it never decreases as the input increases. In other words, the function is either increasing or constant.

For the function f(x) = (x-7)^2, we can see that it is a quadratic function that opens upwards. This means that it is always non-decreasing for all values of x.

To find the largest domain on which f is one-to-one, we need to find the intervals on which the function is strictly increasing. Since the function is a quadratic function, it is strictly increasing on the interval (-∞, 7) and strictly decreasing on the interval (7, ∞).

However, since we are looking for the largest domain on which f is one-to-one and non-decreasing, we can consider the interval (-∞, 7) as the largest domain.

Domain: (-∞, 7)

2. Range of f

To find the range of f, we need to consider the possible values that the function can take.

Since the function is a quadratic function that opens upwards, it has a minimum value at the vertex (7, 0). As we move away from the vertex, the function increases without bound.

Therefore, the range of f is all real numbers greater than or equal to 0.

Range: [0, ∞)

3. Find the Inverse of f

To find the inverse of f, we need to swap the x and y variables and solve for y.

Let's start by writing the function as y = (x-7)^2.

To find the inverse, we can swap the x and y variables to get x = (y-7)^2.

Now, we can solve for y by taking the square root of both sides:

y - 7 = ±√x

y = 7 ± √x

This is the inverse of the function f(x) = (x-7)^2.

Inverse of f: y = 7 ± √x

In conclusion, we have found the largest domain on which f is one-to-one and non-decreasing, the range of f, and the inverse of f.

Key Takeaways

The largest domain on which f is one-to-one and non-decreasing is (-∞, 7).
The range of f is [0, ∞).
The inverse of f is y = 7 ± √x.

Final Thoughts

In this article, we have explored the properties of the function f(x) = (x-7)^2. We have found the largest domain on which f is one-to-one and non-decreasing, the range of f, and the inverse of f. These results provide valuable insights into the behavior of the function and can be used to solve problems involving quadratic functions.
Q&A: Exploring the Properties of the Function f(x) = (x-7)^2

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the function f(x) = (x-7)^2.

Q1: What is the largest domain on which f is one-to-one and non-decreasing?

A1: The largest domain on which f is one-to-one and non-decreasing is (-∞, 7).

Q2: What is the range of f?

A2: The range of f is [0, ∞).

Q3: How do I find the inverse of f?

A3: To find the inverse of f, you can swap the x and y variables and solve for y. The inverse of f is y = 7 ± √x.

Q4: What is the vertex of the parabola represented by f(x) = (x-7)^2?

A4: The vertex of the parabola represented by f(x) = (x-7)^2 is (7, 0).

Q5: Is f(x) = (x-7)^2 an even function?

A5: Yes, f(x) = (x-7)^2 is an even function because f(-x) = f(x) for all x.

Q6: Is f(x) = (x-7)^2 a one-to-one function?

A6: No, f(x) = (x-7)^2 is not a one-to-one function because it is decreasing on the interval (7, ∞).

Q7: How do I determine if a function is one-to-one?

A7: To determine if a function is one-to-one, you can use the horizontal line test. If no horizontal line intersects the graph of the function in more than one place, then the function is one-to-one.

Q8: What is the difference between a one-to-one function and a non-decreasing function?

A8: A one-to-one function is a function that passes the horizontal line test, meaning that no horizontal line intersects the graph of the function in more than one place. A non-decreasing function is a function that never decreases as the input increases.

Q9: Can I use the inverse of f to solve equations involving f?

A9: Yes, you can use the inverse of f to solve equations involving f. To do this, you can substitute the inverse of f into the equation and solve for x.

Q10: How do I graph the function f(x) = (x-7)^2?

A10: To graph the function f(x) = (x-7)^2, you can use a graphing calculator or a computer algebra system. You can also use the vertex form of the parabola to graph the function.

Conclusion

In this article, we have answered some of the most frequently asked questions about the function f(x) = (x-7)^2. We have discussed the largest domain on which f is one-to-one and non-decreasing, the range of f, and the inverse of f. We have also provided answers to other questions about the function, including its vertex, evenness, and one-to-oneness.