Question

If f(x) = max(x, $\frac{1}{x}$) for x > 0 where max (a, b) denotes the greater of the two real numbers a and b. Define the function g(x) = f(x) f($\frac{1}{x}$) and plot its graph.

Answer 1

Given,

$f\left(x\right)=max(x,\frac{1}{x})$

Before we set out to answer the question,we digress to the definition of

$h\left(x\right)=$maximum$\left\{p\right(x),q(x\left)\right\}$

$h\left(x\right)=\{\begin{array}{c}p\left(x\right);whenp\left(x\right)>q\left(x\right)\\ q\left(x\right);whenq\left(x\right)>p\left(x\right)\end{array}$

Thereby, the exact functional form of the given function $f\left(x\right)=max(x,\frac{1}{x})$can be found out if we can break down the mentioned domain into two subintervals , one where $x>\frac{1}{x}$ and the other where $\frac{1}{x}>x$.And this can be determined if we plot both $x\&\frac{1}{x}$ on the same graph.

Hence, the function

$f\left(x\right)=max(x,\frac{1}{x})=\{\begin{array}{c}\frac{1}{x};whenxϵ(0,1]\\ x;whenxϵ[1,\infty )\end{array}$

(this can be understood by assuming the value of $x=\frac{1}{2}or\frac{1}{3}\&2or3$)

Similarly,

$f\left(\frac{1}{2}\right)=max(\frac{1}{x},x)=\{\begin{array}{c}\frac{1}{x};whenxϵ(0,1]\\ x;whenxϵ[1,\infty )\end{array}$

Thereby,

$g\left(x\right)=f\left(x\right).f\left(\frac{1}{x}\right)=\{\begin{array}{c}\frac{1}{x^2};whenxϵ(0,1]\\ x^2;whenxϵ[1,\infty )\end{array}$

Now that we have the functional form of $g\left(x\right)$ , we can plot the graph of $g\left(x\right)$.