Consider the function f(X)= ∛x+4. write an equation that gives the distance from any point on the graph of f(x) to the point (4,-2)

To find the equation for the distance from any point \((x, f(x))\) on the graph of the function \(f(x) = \sqrt[3]{x} + 4\) to the point \((4, -2)\), we use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

In this case:
- The first point is any point \((x, f(x))\) on the graph of \(f(x)\), so the coordinates are \((x, \sqrt[3]{x} + 4)\).
- The second point is \((4, -2)\).

Substituting into the distance formula:

\[
D = \sqrt{(x - 4)^2 + \left( \left( \sqrt[3]{x} + 4 \right) - (-2) \right)^2}
\]

Simplify the expression:

\[
D = \sqrt{(x - 4)^2 + \left( \sqrt[3]{x} + 4 + 2 \right)^2}
\]

\[
D = \sqrt{(x - 4)^2 + \left( \sqrt[3]{x} + 6 \right)^2}
\]

Thus, the equation that gives the distance from any point on the graph of \(f(x) = \sqrt[3]{x} + 4\) to the point \((4, -2)\) is:

\[
D = \sqrt{(x - 4)^2 + \left( \sqrt[3]{x} + 6 \right)^2}
\]

Would you like further details on this equation, or do you have any other questions?

Here are 5 related questions to expand your understanding:
1. How would you compute the distance for a specific value of \(x\)?
2. Can you find the minimum distance from the graph of \(f(x)\) to the point \((4, -2)\)?
3. What happens to the distance \(D\) as \(x\) increases or decreases?
4. How would you graph the function \(f(x) = \sqrt[3]{x} + 4\) and plot the point \((4, -2)\)?
5. What is the geometrical significance of minimizing the distance from a curve to a point?

**Tip:** Always remember to simplify expressions involving square roots and cube roots, especially when applying them in formulas like the distance formula.

Consider the function f(X)= ∛x+4. Write an equation that gives the distance from any point on the graph of f(x) to the point (4,-2).

Learn how to calculate the distance from any point on the graph of f(x) = ∛x + 4 to a specific point (4, -2) using the distance formula. This problem covers algebraic concepts such as the cube root function and distance between points. Ideal for high school students in Grades 9-12.

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