Domain & Range Of F(x) = √(x-7) + 9 Explained!

Nov 13, 2025 by Andrew McMorgan 47 views

What are the domain and range of the function f(x)=√(x-7)+9?

Hey guys! Let's break down how to find the domain and range of the function f(x) = √(x-7) + 9. This is a super common type of problem in algebra and calculus, and once you understand the basics, it becomes pretty straightforward. We'll go step-by-step to make sure you get it.

Understanding Domain and Range

Before diving into this specific function, let's quickly recap what domain and range mean.

Domain: The domain of a function is the set of all possible input values (x-values) that the function can accept without causing any mathematical errors. Essentially, it's all the x-values you can plug into the function.
Range: The range of a function is the set of all possible output values (y-values) that the function can produce. It represents all the possible results you can get from the function.

Why Domain Matters

The domain is crucial because some operations are not defined for certain numbers. For example:

You can't take the square root of a negative number (in the real number system).
You can't divide by zero.
You can't take the logarithm of a non-positive number.

When finding the domain, we want to avoid these undefined operations. For the given function, the square root is the main constraint we need to consider.

Why Range Matters

The range tells us what values our function can actually output. Understanding the range can help us analyze the behavior of the function and its possible values. It gives us insight into the function’s limitations and capabilities.

Finding the Domain of f(x) = √(x-7) + 9

The function we're working with is f(x) = √(x-7) + 9. The most important part here is the square root, √(x-7). Remember, we can't take the square root of a negative number (in the real number system). So, the expression inside the square root, (x-7), must be greater than or equal to zero.

Setting up the Inequality

To find the domain, we set up the following inequality:

x - 7 ≥ 0

Solving for x

Now, we solve for x:

x ≥ 7

Interpreting the Result

This inequality tells us that x must be greater than or equal to 7. In other words, the smallest value x can be is 7, and it can be any number larger than that. This ensures that the expression inside the square root is non-negative.

Writing the Domain

We can write the domain in a few different ways:

Inequality Notation: x ≥ 7
Interval Notation: [7, ∞)

Both notations mean the same thing: x can be any number from 7 (inclusive) to infinity.

Finding the Range of f(x) = √(x-7) + 9

Now that we've found the domain, let's tackle the range. The function is f(x) = √(x-7) + 9. We know that the square root part, √(x-7), will always be non-negative because the square root of a real number is always greater than or equal to zero.

Analyzing the Square Root Part

Since √(x-7) is always non-negative, its minimum value is 0. This happens when x = 7:

√(7-7) = √0 = 0

Considering the + 9

The function adds 9 to the square root, so the entire function will always be greater than or equal to 9. The smallest value f(x) can be is when √(x-7) is 0:

f(x) = 0 + 9 = 9

Determining the Range

As x gets larger, √(x-7) also gets larger, and so does f(x). There's no upper limit to how large x can be, so there's no upper limit to how large f(x) can be.

Writing the Range

We can write the range in a few different ways:

Inequality Notation: y ≥ 9
Interval Notation: [9, ∞)

Both notations mean the same thing: y can be any number from 9 (inclusive) to infinity.

Putting It All Together

So, for the function f(x) = √(x-7) + 9:

Domain: x ≥ 7 or [7, ∞)
Range: y ≥ 9 or [9, ∞)

Looking at the options provided:

A. domain: x ≥ -7, range: y ≥ 9 (Incorrect) B. domain: x ≥ 7, range: y ≥ -9 (Incorrect) C. domain: x ≥ 7, range: y ≥ 9 (Correct) D. domain: x ≥ 9, range: y ≥ 7 (Incorrect)

The correct answer is C. The domain is x ≥ 7, and the range is y ≥ 9.

Visualizing the Function

Graphing the function can also help you visualize the domain and range. The graph of f(x) = √(x-7) + 9 starts at the point (7, 9) and extends upwards and to the right. This visually confirms that the domain is x ≥ 7 and the range is y ≥ 9.

Common Mistakes to Avoid

Forgetting about the square root: Always remember that the expression inside a square root must be non-negative.
Incorrectly solving the inequality: Make sure you solve the inequality x - 7 ≥ 0 correctly.
Mixing up domain and range: Keep in mind that the domain refers to the x-values, and the range refers to the y-values.
Ignoring the vertical shift: Don't forget to consider the + 9, which shifts the entire function upwards and affects the range.

Practice Problems

To solidify your understanding, try finding the domain and range of these functions:

g(x) = √(x - 5) + 3
h(x) = √(x + 2) - 1
k(x) = √(2x - 4) + 6

Conclusion

Finding the domain and range of a function like f(x) = √(x-7) + 9 involves understanding the constraints imposed by the square root and considering any vertical shifts. By setting up and solving inequalities, and by analyzing the behavior of the function, you can accurately determine the domain and range. Keep practicing, and you'll master these concepts in no time! Remember, the domain is all about what x can be, and the range is all about what y can be. You got this!