Solved: What does this sentence mean: Geometrically, a root of $(1)$ is that value of $~x~$ when the graph of $~f(x)~$ crosses the $~x$-axis?

Question

Thanks Quasi for solving my question, what i really meant to ask is what the author means by root of (1) and the answer to that is that it is just a label for the equation and the same thing as say for a root of f(x).

This sentence is from the book Higher Engineering Mathematics by B.S. Grewal

What does this sentence mean: Geometrically, a root of $(1)$ is that value of $~x~$ when the graph of $~f(x)~$ crosses the $~x$-axis ?

Answer 1

If $r\in\mathbb{R}$, then $r$ is a root of the equation $f(x)=0$

$\iff\;f(r)=0$

$\iff\;$the equation $y=f(x)$ is such that $y=0$ when $x=r$

$\iff\;$the point $(r,0)$ is a point on the graph of the equation $y=f(x)$

$\iff\;$the graph of the equation $y=f(x)$ intersects the $x$-axis at $x=r$

Note:

The author was a little careless when using the phrase "crosses the $x$-axis".

Correct that to "intersects the $x$-axis".

Answer 2

Consider your $f$ as a function $f:\mathbb{R}\to \mathbb{R}$, so the graph of $f$ is a subset of the plane $\mathbb{R}^2$. Then a root of $f$ is a point $x\in \mathbb{R}$ such that $f(x)=0$, and so the point $(x,f(x))=(x,0)$ lies on the $x$-axis in the plane $\mathbb{R}^2$.