Question

Graph the functions \(f(x)=\ln x\) and \(g(x)=x^{1 / 4}\) in the same viewing window. Where do these graphs intersect? As \(x\) increases, which function grows more rapidly?

Short answer

The graphs intersect approximately at \(x = 1\) and \(x \approx 2.5\). As \(x\) increases, the function \(g(x) = x^{1/4}\) grows more rapidly.

Step-by-step solution

Graph \(f(x) = \ln x\) and \(g(x) = x^{1/4}\)

For the first step, graph the two functions within the same viewing window. Both functions only accept positive \(x\), so ensure to choose a suitable range, for instance from \(x = 0\) to \(x = 5\).

Identify Intersection Points

After graphing the two functions, find out where \(f(x)\) equals \(g(x)\). Equate the two functions and solve for \(x\), i.e., \(\ln x = x^{1/4}\). Since it is not straightforward to solve the equation algebraically, a numerical method could be used, such as the bisection method, or you could estimate the value by closely looking at the graph.

Determine Which Function Grows More Rapidly

Lastly, determine which function grows more rapidly as \(x\) increases. An easier way to do this is by taking the limit of the ratio of the two functions as \(x\) approaches infinity, namely, \(\lim_{{x \to \infty}} \frac{\ln x}{x^{1/4}}\) . If this limit is 0, \(g(x)\) grows faster; if it is infinity, then \(f(x)\) grows faster; and if it is a constant, both functions grow at the same rate.