Question

How do you sketch $f\left( x \right)={{\cos }^{3}}x$?

Answer 1

Hint: We first explain the cubic curve for the trigonometric ratio $f\left( x \right)={{\cos }^{3}}x$. Then we place the values and different signs for $x$ and $y$ coordinates in the function $f\left( x \right)={{\cos }^{3}}x$. We tried to find the characteristics for the graph and then plot the graph. We also find the range of the function.

Complete step by step answer:
The given equation of $f\left( x \right)={{\cos }^{3}}x$ is an example of rectangular hyperbola.
The given equation graph will be the cubic version of $y=\cos x$. We know that for the function $y=\cos x$, the range is always $\left[ -1,1 \right]$.
In the case of $f\left( x \right)={{\cos }^{3}}x$, the range still remains $\left[ -1,1 \right]$.
We can find the graph using the values separately.
We take two functions $y=\cos x$ and $f\left( x \right)={{\cos }^{3}}x$ together to compare the values.
We take $y=\cos x<0$ which gives $f\left( x \right)={{\cos }^{3}}x<0$. Similarly, if we take $y=\cos x>0$, then that will give $f\left( x \right)={{\cos }^{3}}x>0$.
If the value of $y=\cos x$ increases, the value of $f\left( x \right)={{\cos }^{3}}x$ decreases because of the range value being in $\left[ -1,1 \right]$. Similarly, if the value of $y=\cos x$ decreases, the value of $f\left( x \right)={{\cos }^{3}}x$ increases.
This above-mentioned relation of $x$ and $y$ don’t work for points $y=f\left( x \right)=-1,0,1$
Now we based on the information, draw the graph.

Note: We need to remember that the function $f\left( x \right)={{\cos }^{3}}x$ is continuous at all points.
The value of $f\left( x \right)$ in the function $f\left( x \right)={{\cos }^{3}}x$ changes rapidly in the interval $\left( -1,1 \right)\backslash \left\{ 0 \right\}$.