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Domain, Range and Comparing Common Functions - Answers

These are the answers to the questions in the tutorial: Graph, Domain and Range of Common Functions.

TUTORIAL (1) - Domain and Range of Basic Functions

  1. linear function f(x) = x

    domain : (- ∞ , + ∞)

    range : (- ∞ , + ∞)

  2. squaring function g(x) = x2

    domain : (- ∞ , + ∞)

    range : [0 , + ∞)

  3. cubing function h(x) = x3

    domain : (- ∞ , + ∞)

    range : (- ∞ , + ∞)

  4. absolute value function i(x) = |x|

    domain : (- ∞ , + ∞)

    range : [0 , + ∞)

  5. square root function j(x) = √(x)

    domain : [0 , + ∞)

    range : [0 , + ∞)

  6. cube root function k(x) = 3√(x)

    domain : (- ∞ , + ∞)

    range : (- ∞ , + ∞)

  7. natural exponential function l(x) = e x

    domain : (- ∞ , + ∞)

    range : (0, + ∞)

  8. natural logarithmic function m(x) = ln(x)

    domain : (0 , + ∞)

    range : (- ∞, + ∞)

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 x3 < x2 < x < √(x) < 3√(x)

for x > 1 we have the inequality x3 > x2 > 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) = x2 ; decreasing over the interval (-∞ , 0) , increasing over the interval ( 0 , ∞) , local minimum at (0,0).

3 ) h(x) = x3 ; 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) = ex ; 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) = x2. (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) = x2 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) = x3 and k(x) = 3√ x because:

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

h(- x) = (- x)3 = - x3 = - 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) = ex and m(x) = ln x are neither even nor odd.