Special Functions

The integral  ${\int }_0^{1.5}\left[x^2\right]dx,$ where $\left[.\right]$ denotes the greatest integer function, equals:

If $f\left(x\right)=\sec x+\cos x$ and $g\left(x\right)=\mathrm{cosec}x+\sin x$, show that$f\left(x\right)$ is an even function and $g\left(x\right)$ is an odd function of $x$.

Find if the following function is an even or an odd function.

Find if the following function is an even or an odd function.

Find if the following function is an even or an odd function.

$x+\sin x$

If $f\left(x\right)=x+3x^3$ and $g\left(x\right)=3-x^2$, show that$f\left(x\right)$ is an odd function and $g\left(x\right)$ is an even function of $x$.

Solve $3\left|5-2x\right|>7$

Show that the function $f\left(2x\right)=2x$ is an identity function.

Evaluate the ceiling function of $4.5$ and $-4.5$. Also, explain the answer.

Define the smallest integer function or ceiling function. Find the range of ceiling function by using graph of ceiling function.

Define the smallest integer function or ceiling function. Find the domain of ceiling function by using graph of ceiling function.

Draw the graph of the smallest integer function or ceiling function.

Define the smallest integer function or ceiling function with one example.

Define greatest integer function. Also draw its graph and write the domain and range of this function.

Draw the graph of step function $y=\left[x\right]$ and find the value of $\left[-3.2\right]$.

Draw the graph of step function $y=\left[x\right]$ and find the value of $\left[-1\right]$.

Draw the graph of step function $y=\left[x\right]$ and find the value of $\left[-1.99\right]$.

Draw the graph of step function $y=\left[x\right]$ and find the value of $\left[1.99\right]$.

Draw the graph of step function $y=\left[x\right]$ and find the value of $\left[2.2\right]$.

$f\left(x\right)=4\frac{\left|x\right|}{x}$. State whether given function is even or odd.