take f(x)= square root of x and V=<-3,-6>. Identify a formula for g, where g=V+f

Given that \( f(x) = \sqrt{x} \) and \( V = \langle -3, -6 \rangle \), we need to identify a formula for \( g(x) \), where \( g(x) = V + f(x) \).

Here's how we can define it step by step:

1. \( f(x) = \sqrt{x} \) is a scalar function.
2. \( V = \langle -3, -6 \rangle \) is a vector.
3. To add the vector \( V \) to the function \( f(x) \), we'll interpret this as adding the components of \( V \) to the result of \( f(x) \).

So, \( g(x) = \langle g_1(x), g_2(x) \rangle \), where:
- \( g_1(x) = -3 + \sqrt{x} \)
- \( g_2(x) = -6 + \sqrt{x} \)

Thus, the formula for \( g(x) \) is:

\[
g(x) = \langle -3 + \sqrt{x}, -6 + \sqrt{x} \rangle
\]

This represents a vector-valued function where each component is shifted by the components of \( V \).

Would you like more details or clarification on any part of the process?

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**Related Questions:**
1. What happens to \( g(x) \) as \( x \) approaches 0?
2. How would the formula change if \( V = \langle a, b \rangle \) instead of \( \langle -3, -6 \rangle \)?
3. How would \( g(x) \) behave if \( f(x) = x^2 \) instead of \( \sqrt{x} \)?
4. Can you find the derivative of \( g(x) \) for each component?
5. How do the transformations in \( g(x) \) affect the graph of \( f(x) \)?

**Tip:** Vector-valued functions like \( g(x) \) can be thought of as defining a path in the plane where each component affects one dimension of the motion.

Take f(x) = square root of x and V = <-3, -6>. Identify a formula for g, where g = V + f.

Learn how to define the vector-valued function g(x) when f(x) = sqrt(x) and V = <-3, -6>. This problem covers algebraic concepts and the manipulation of vector-valued functions, suitable for high school students.

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