1. **State the problem:** We are given the function $f(x) = 4x - \frac{1}{x}$ and asked to find $f(x + \Delta x)$. 2. **Recall the formula:** To find $f(x + \Delta x)$, substitute $x + \Delta x$ into the function in place of $x$: $$f(x + \Delta x) = 4(x + \Delta x) - \frac{1}{x + \Delta x}$$ 3. **Substitute and simplify:** $$f(x + \Delta x) = 4x + 4\Delta x - \frac{1}{x + \Delta x}$$ 4. **Explain:** This expression shows the value of the function at $x + \Delta x$. It consists of a linear term $4x + 4\Delta x$ and a rational term $-\frac{1}{x + \Delta x}$. 5. **Note on the parabola:** The function $x^2 - 1$ describes a parabola, but it is separate from the function $f(x)$ given. The parabola is located near the bottom half of the page as per the description, but it is not directly related to the calculation of $f(x + \Delta x)$. **Final answer:** $$f(x + \Delta x) = 4x + 4\Delta x - \frac{1}{x + \Delta x}$$