Math Problem Statement

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

Solution

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

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

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

So, g(x)=g1(x),g2(x)g(x) = \langle g_1(x), g_2(x) \rangle, where:

  • g1(x)=3+xg_1(x) = -3 + \sqrt{x}
  • g2(x)=6+xg_2(x) = -6 + \sqrt{x}

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

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

This represents a vector-valued function where each component is shifted by the components of VV.

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


Related Questions:

  1. What happens to g(x)g(x) as xx approaches 0?
  2. How would the formula change if V=a,bV = \langle a, b \rangle instead of 3,6\langle -3, -6 \rangle?
  3. How would g(x)g(x) behave if f(x)=x2f(x) = x^2 instead of x\sqrt{x}?
  4. Can you find the derivative of g(x)g(x) for each component?
  5. How do the transformations in g(x)g(x) affect the graph of f(x)f(x)?

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

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Math Problem Analysis

Mathematical Concepts

Vector-Valued Functions
Algebra
Square Roots

Formulas

g(x) = <-3 + sqrt(x), -6 + sqrt(x)>

Theorems

-

Suitable Grade Level

Grades 9-12