Functions in Algebra Examples

Determine whether the given set of functions is linearly dependent or linearly independent on the interval ${\displaystyle (-\mathrm{\infty },\mathrm{\infty })}$
${\displaystyle f_1\left(x\right)=1+x,}$

$f_2\left(x\right)=x,$

$f_3\left(x\right)=x^2$

$\text{If X is uniformly distributed over }(-1,1),\text{ find (a) }P|X|>\frac{1}{2}$
(b) the density function of the random variable |X|.

In order for the function to have the specified maximum or minimum value, find the values of b.
${\displaystyle f\left(x\right)=-x^2+bx-75}$ 
Maximum value: 25

For what value of the constant c is the function f continuous on $(-\mathrm{\infty },\mathrm{\infty })$?
$f(x)=\frac{c{x}^2+2x}{x^2-1}$ if $x<2$ and $x^3-cx$ if $x\ge 2$

Find a function f such that ${\displaystyle F=▽}$
${\displaystyle F(x,y)=x^{2}i+y^{2}j}$
C is the arc of the parabola ${\displaystyle y=2x^2}$ from (-1,2) to (2,8)

Suppose that $f(x)=0.25for0<x<4$. Calculate X's mean and variance.

Whether each of these functions is a bijection from R to R.

a) $f(x)=-3x+4$

b) ${\displaystyle f\left(x\right)=-3x^2+7}$

c) $f(x)=\frac{x+1}{x+2}$
${\displaystyle d)f\left(x\right)=x^5+1}$