__TUTORIAL (2) - Comparing Basic Functions__

1 ) For x ≥ 0, | x | = x, hence the same graphs for f(x) = x and i(x) = | x | for x ≥ 0. For x < 0, | x | = - x > 0 which explains why the graph of i (x) = | x | is above the x axis for x < 0.

2 ) l(x) = e^{ x} and m(x) = ln (x) are inverse of each other, hence the graphs are reflections of each other on the line y = x.

3 ) h(x) = x^{ 3} and k(x) = ^{3}√(x) are inverse of each other, hence the graphs are reflections of each other on the line y = x.

4 ) for 0 < x < 1 we have the inequality x^{3} < x^{2} < x < √(x) < ^{3}√(x)

for x > 1 we have the inequality x^{3} > x^{2} > x > √(x) > ^{3}√(x)

__TUTORIAL (3) - Intervals of Increase and Decrease and any local minimum or maximum of the Basic Functions__

1 ) f(x) = x ; increasing over the interval (-∞ , +∞) , no local minimum or maximum.

2 ) g(x) = x^{2} ; decreasing over the interval (-∞ , 0) , increasing over the interval ( 0 , ∞) , local minimum at (0,0).

3 ) h(x) = x^{3} ; increasing over the interval (-∞ , +∞) , no local minimum or maximum.

4 ) i(x) = | x | ; decreasing over the interval (-∞ , 0) , increasing over the interval ( 0 , ∞) , local minimum at (0,0).

5 ) j(x) = √ x increasing over the interval [ 0 , +∞) , local minimum at (0,0).

6 ) k(x) = ^{3}√ x ; increasing over the interval (-∞ , +∞) , no local minimum or maximum.

7 ) l(x) = e^{x} ; increasing over the interval (-∞ , +∞) , no local minimum or maximum.

8 ) m(x) = ln x ; increasing over the interval (-∞ , +∞) , no local minimum or maximum.

__TUTORIAL (4) - Compare the rate of change of the Basic Functions__

1 ) As x increases, l(x) = e^{ x} increases faster than g(x) = x^{2}. (Graph and compare)

2 ) As x increases, f(x) = x increases faster than m(x) = ln x. (Graph and compare)

__TUTORIAL (5) - Identify Even and Odd Basic Functions__

1 ) Even basic functions are: g(x) = x^{2} and i(x) = | x | because:

g(-x) = (-x)^{2} = x ^{2} = g(x)

and i( - x) = | - x | = | x | = i(x).

Also, if you graph functions g and i, their graphs are symmeteric with respect to the y axis.

2 ) Odd basic functions are: f(x) = x , h(x) = x^{3} and k(x) = ^{3}√ x because:

f(-x) = - x = - f(x)

h(- x) = (- x)^{3} = - x^{3} = - h(x)

k( - x) = ^{3}√ (- x) = - ^{3}√x = - k(x).

Also if you graph functions f, h and k, their graphs are symmeteric with respect to the origin of the system of axes.

3 ) Functions j(x) = √x , l(x) = e^{x} and m(x) = ln x are neither even nor odd.