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When conducting a graphical analysis, one of the effective ways to understand how two functions behave in relation to each other is by plotting them on a graph. In this case, we are comparing the functions \(f(x) = x\) and \(g(x) = \tan x\) within the interval \( \(0, \pi / 2\)\). By graphing these two functions, we can visually examine which function is greater.The function \(f(x) = x\) is a straight line through the origin with a consistent increase as \(x\) increases. Meanwhile, \(g(x) = \tan x\) is a curve that starts at the origin but rapidly increases as \(x\) approaches \(\pi/2\).

• \(f(x) = x\) is a linear function.
• \(g(x) = \tan x\) increases exponentially as it nears \(\pi/2\).

This difference in growth rates means that on a graph, \(g(x)\) will sit above \(f(x)\), visually demonstrating that \(g(x) > f(x)\) on the interval \((0, \pi / 2)\). This powerful visual tool helps us confirm our numerical observations and make confident conjectures about the behavior of these functions.

An analytic proof uses mathematical reasoning to demonstrate the inequalities between functions. In our exercise, we need to show that \(f(x) = x\) is less than \(g(x) = \tan x\) for the interval \((0, \pi/2)\).To establish this inequality analytically, we define a new function \(h(x) = g(x) - f(x) = \tan x - x\). The task is to determine if \(h(x)\) is greater than zero for all \(x\) in the given interval. To do this, calculate the derivative of \(h(x)\): \[ h'(x) = \frac{d}{dx}(\tan x) - \frac{d}{dx}(x) = \sec^2 x - 1.\]Since \(\sec^2 x\) is always greater than 1 for \(xeq 0\) in the interval \((0, \pi / 2)\), we can see that \(h'(x) > 0\). A positive derivative indicates that \(h(x)\) is increasing, and hence \(\tan x > x\) throughout the interval. By showing this analytically, we demonstrate that \(f(x)\) is indeed less than \(g(x)\) over the specified interval, confirming our graphical analysis findings.

When comparing functions, it’s important to consider various criteria to fully understand how they interact and relate. For the functions \(f(x) = x\) and \(g(x) = \tan x\), we explore numerical calculation, graphical representation, and analytic proof.Numerically, evaluate these functions at different points within the interval \((0, \pi/2)\):

• \(f(0.25) = 0.25, g(0.25) = \tan(0.25)\), compare values.
• Continue for \(x = 0.5, 0.75, 1, 1.25, 1.5\).

Observed values will typically show that \(g(x)\) is greater than \(f(x)\).Graphically, the curve for \(\tan x\) rises steeply compared to the linear increase of \(x\), reinforcing that \(g(x)\) surpasses \(f(x)\).Analytically, we proved \(\tan x > x\) through the positive derivative \(h'(x) = \sec^2 x - 1\). This method confirms supremity across the interval.Through these methods, understanding becomes clearer. Each approach corroborates the others, forming a comprehensive picture of how these functions behave compared to one another.