1. **State the problem:** We are given the function $$f(x) = e^x \sqrt[3]{3x t \sin x}$$ and we want to understand or analyze it. 2. **Recall the components:** - The function involves an exponential term $$e^x$$. - It also involves a cube root $$\sqrt[3]{3x t \sin x}$$, which means the cube root of the product of 3, x, t, and $$\sin x$$. 3. **Important notes:** - The exponential function $$e^x$$ is always positive for all real $$x$$. - The cube root function $$\sqrt[3]{y}$$ is defined for all real numbers $$y$$, including negative values. - The product inside the cube root is $$3x t \sin x$$, so the sign depends on $$x$$, $$t$$, and $$\sin x$$. 4. **Simplify or rewrite the function:** $$f(x) = e^x \left(3x t \sin x\right)^{\frac{1}{3}}$$ 5. **Interpretation:** - The function grows exponentially with $$x$$ due to $$e^x$$. - The cube root moderates the growth or decay depending on the values of $$x$$, $$t$$, and $$\sin x$$. 6. **If needed, to evaluate or plot, substitute values for $$t$$ and $$x$$.** Final expression: $$f(x) = e^x \left(3x t \sin x\right)^{\frac{1}{3}}$$