Find a domain on which the function *f*(*x*)=(*x*−7)^2

 is one-to-one and non-decreasing.
 
Write the domain in interval notation.
 
To enter ∞

, type infinity. To enter ∪, type U.

To find a domain where the function \( f(x) = (x - 7)^2 \) is one-to-one and non-decreasing, we need to understand the behavior of this quadratic function.

### Step 1: Analyze the function
The function \( f(x) = (x - 7)^2 \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive). Its vertex is at \( x = 7 \), and the function decreases for \( x < 7 \) and increases for \( x > 7 \).

### Step 2: Restrict the domain
To make the function one-to-one and non-decreasing, we need to restrict the domain to the right of the vertex, where the function is non-decreasing. This means we take the domain from \( x = 7 \) to \( \infty \).

### Step 3: Write the domain
The domain on which the function is one-to-one and non-decreasing is:
\[
[7, \infty)
\]

Would you like more details, or do you have any questions?

Here are some related questions to further your understanding:
1. How do you determine if a function is one-to-one?
2. What is the range of the function \( f(x) = (x - 7)^2 \) on the domain \( [7, \infty) \)?
3. Why do parabolas have a vertex, and how does it relate to the function being one-to-one?
4. Can the function \( f(x) = (x - 7)^2 \) be one-to-one if we restrict the domain to \( (-\infty, 7

Find a domain on which the function f(x) = (x − 7)^2 is one-to-one and non-decreasing. Write the domain in interval notation.

Discover how to find a domain where the function f(x) = (x - 7)^2 is one-to-one and non-decreasing. This solution explores key concepts in quadratic functions and monotonicity, suitable for students in Grades 10-12.

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