Root hunter

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

Take a look at the function in the graph below.

Image
Root hunter

The graph is positive for $x = 5$ and negative for $x = 3$. This means that the graph must cut the $x$ axis somewhere between $3$ and $5$.

Although in this case the result is obvious (because we have the whole graph to look at!), we can also use this idea to show that more tricky functions also have roots.

Use this idea to show that these functions possess at least one solution $f(x) = 0$:

$$ f(x)=\frac{1}{x-2}+\frac{1}{x-3} $$ $$f(x)= x^x - 1.5 x$$ $$f(x)= x^{1000000}+{1000000}^x - 17$$ $$f(x)=\cos(\sin(\cos x)) - \sin(\cos(\sin x)) $$

Optional extension activity: Can you make a spreadsheet that helps you find the numerical values of the roots to, say, four decimal places?