As an entry-level financial analyst, you are asked to prepare a visual report comparing three different compounding growth projections for a corporate savings fund over time $x$ (for positive values of $x$). The three projections are modeled by the functions $g(x) = 2^x$, $h(x) = 3^x$, and the natural exponential function $f(x) = e^x$. Based on the mathematical value of the natural base $e$, what do you recall about the position of the graph of $f(x) = e^x$ relative to the other two projections on your chart?

An engineering technician is comparing three different exponential growth models for a system's energy output over time. The models are defined by the functions $g(x) = 2^x$, $f(x) = e^x$, and $h(x) = 3^x$. Match each function with the description that correctly identifies its relative steepness and position for positive values of $x$ (where $x > 0$).

A business analyst is comparing three exponential growth models to project future sales over time $x$ (where $x > 0$): Model A ($g(x) = 2^x$), Model B ($h(x) = 3^x$), and Model C ($f(x) = e^x$). To ensure the models are plotted correctly on a growth chart, the analyst must recall their relative steepness. Arrange the following models in order from the slowest growth (least steep) to the fastest growth (most steep).

A financial analyst is comparing three revenue growth models for a corporate report: $g(x) = 2^x$, $f(x) = e^x$, and $h(x) = 3^x$. For all positive values of $x$, the analyst correctly identifies that the graph of $f(x) = e^x$ will lie precisely between the graphs of $g(x) = 2^x$ and $h(x) = 3^x$.

Graphical Position of the Base e

Comparing Website Traffic Growth Models

An inventory planner at a hardware supply company is comparing three different growth models to project stock demand over positive time intervals $x$. The models are defined by the functions $g(x) = 2^x$, $h(x) = 3^x$, and the natural exponential function $f(x) = e^x$. To help warehouse staff visualize the three projections on a shared chart, the planner explains that because the natural base $e$ is approximately 2.718, the curve representing $f(x) = e^x$ will lie precisely ____ the curves representing $g(x) = 2^x$ and $h(x) = 3^x$ for positive values of $x$.