To complete the task for the function
F
(
x
)
=
x
+
3
F(x)=
x
​
+3, we'll evaluate this function for integer values of
x
x that lead to whole numbers under the square root. This will ensure there are no decimal values in the results for
y
=
F
(
x
)
y=F(x).

To complete the task for the function \( F(x) = \sqrt{x + 3} \), we need to evaluate this function for integer values of \( x \) that ensure \( x + 3 \) is a perfect square. This will avoid decimal results in \( y = F(x) \).

### Step-by-Step Process:
1. For \( y = F(x) = \sqrt{x + 3} \) to be a whole number, \( x + 3 \) must be a perfect square. That is:
   \[
   x + 3 = n^2
   \]
   where \( n \) is a whole number.

2. Solving for \( x \), we get:
   \[
   x = n^2 - 3
   \]
   Now, we'll evaluate this expression for different values of \( n \).

### Example Calculations:
- When \( n = 2 \): 
  \[
  x = 2^2 - 3 = 4 - 3 = 1
  \]
  So, \( F(1) = \sqrt{1 + 3} = \sqrt{4} = 2 \).

- When \( n = 3 \):
  \[
  x = 3^2 - 3 = 9 - 3 = 6
  \]
  So, \( F(6) = \sqrt{6 + 3} = \sqrt{9} = 3 \).

- When \( n = 4 \):
  \[
  x = 4^2 - 3 = 16 - 3 = 13
  \]
  So, \( F(13) = \sqrt{13 + 3} = \sqrt{16} = 4 \).

### General Expression:
Thus, for any \( n \), the function \( F(x) \) will return a whole number if \( x = n^2 - 3 \).

Do you have any questions about these steps or need further details?

---

Here are 5 related questions to deepen your understanding:
1. What happens if \( x + 3 \) is not a perfect square? How does it affect \( F(x) \)?
2. How can we generalize this approach to functions involving square roots and different constants?
3. What is the domain of the function \( F(x) = \sqrt{x + 3} \), and why does it matter?
4. Can we apply the same logic for cube roots or higher roots?
5. What would the graph of \( F(x) = \sqrt{x + 3} \) look like?

**Tip**: Always check the domain of functions involving square roots to avoid non-real results.

To complete the task for the function F(x) = sqrt(x + 3), we'll evaluate this function for integer values of x that lead to whole numbers under the square root. This will ensure there are no decimal values in the results for y = F(x).

Explore how to evaluate the function F(x) = sqrt(x + 3) by identifying integer values of x that result in whole numbers. Learn the step-by-step process and apply key concepts such as perfect squares and square roots. Suitable for Grades 9-10 students.

Math Problem Statement

To complete the task for the function F ( x )

x + 3 F(x)= x +3, we'll evaluate this function for integer values of x x that lead to whole numbers under the square root. This will ensure there are no decimal values in the results for y

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation