To find the $x$-coordinate of the point on the graph of the function $f(x) = 2x + 3$ that is closest to the origin, we need to minimize the distance from any point on the graph to the origin.

Step 1: Distance Formula

The distance $d$ from any point $(x, f(x))$ on the graph to the origin $(0, 0)$ is given by the distance formula: $d(x) = \sqrt{x^2 + f(x)^2}$ Substitute $f(x) = 2x + 3$: $d(x) = \sqrt{x^2 + (2x + 3)^2}$

Step 2: Minimize the Distance

Since minimizing the square root function will also minimize the function inside the square root, we first minimize $d^2(x)$: $d^2(x) = x^2 + (2x + 3)^2$ Expand the square: $d^2(x) = x^2 + (4x^2 + 12x + 9)$ Simplify: $d^2(x) = 5x^2 + 12x + 9$

Step 3: Take the Derivative

To find the minimum, take the derivative of $d^2(x)$ with respect to $x$: $\frac{d}{dx}[5x^2 + 12x + 9] = 10x + 12$ Set the derivative equal to zero to find the critical points: $10x + 12 = 0$ Solve for $x$: $x = -\frac{12}{10} = -\frac{6}{5}$

Step 4: Verify Minimum

To confirm that this value of $x$ corresponds to a minimum, we check the second derivative: $\frac{d^2}{dx^2}[5x^2 + 12x + 9] = 10$ Since the second derivative is positive, the function is concave up, confirming a minimum at $x = -\frac{6}{5}$.

Thus, the $x$-coordinate of the point on the graph closest to the origin is: $\boxed{-\frac{6}{5}}$

Would you like further details on the solution or have any questions about the process? Here are some related questions to explore:

1. How does the distance formula change if the point isn't the origin?
2. How can you use derivatives to find the closest point on other types of curves?
3. What other methods exist for finding the closest point between a curve and a point?
4. How do you compute the distance between two arbitrary points in space?
5. What happens if the function is nonlinear or not a straight line?

Tip: When minimizing a function that involves a square root, it’s often easier to minimize the square of the distance instead.