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: \( (-\infty, +\infty) \)
   Range: \( (-\infty, +\infty) \)
2. 
   Squaring function \( g(x) = x^2 \)
   Domain: \( (-\infty, +\infty) \)
   Range: \( [0, +\infty) \)
3. 
   Cubing function \( h(x) = x^3 \)
   Domain: \( (-\infty, +\infty) \)
   Range: \( (-\infty, +\infty) \)
4. 
   Absolute value function \( i(x) = |x| \)
   Domain: \( (-\infty, +\infty) \)
   Range: \( [0, +\infty) \)
5. 
   Square root function \( j(x) = \sqrt{x} \)
   Domain: \( [0, +\infty) \)
   Range: \( [0, +\infty) \)
6. 
   Cube root function \( k(x) = \sqrt[3]{x} \)
   Domain: \( (-\infty, +\infty) \)
   Range: \( (-\infty, +\infty) \)
7. 
   Natural exponential function \( l(x) = e^x \)
   Domain: \( (-\infty, +\infty) \)
   Range: \( (0, +\infty) \)
8. 
   Natural logarithmic function \( m(x) = \ln(x) \)
   Domain: \( (0, +\infty) \)
   Range: \( (-\infty, +\infty) \)

Tutorial (2) – Comparing Basic Functions

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

2) The functions \( l(x) = e^x \) and \( m(x) = \ln(x) \) are inverses of each other. Therefore, their graphs are reflections across the line \[ y = x. \]

3) The functions \( h(x) = x^3 \) and \( k(x) = \sqrt[3]{x} \) are inverses of each other. Hence, their graphs are reflections across the line \[ y = x. \]

4) For \( 0 < x < 1 \), we have: \[ x^3 < x^2 < x < \sqrt{x} < \sqrt[3]{x} \] For \( x > 1 \), we have: \[ x^3 > x^2 > x > \sqrt{x} > \sqrt[3]{x} \]

Tutorial (3) – Intervals of Increase/Decrease and Local Extrema

1) \( f(x) = x \): increasing on \( (-\infty, +\infty) \); no local minimum or maximum.

2) \( g(x) = x^2 \): decreasing on \( (-\infty, 0) \), increasing on \( (0, +\infty) \); local minimum at \( (0,0) \).

3) \( h(x) = x^3 \): increasing on \( (-\infty, +\infty) \); no local minimum or maximum.

4) \( i(x) = |x| \): decreasing on \( (-\infty, 0) \), increasing on \( (0, +\infty) \); local minimum at \( (0,0) \).

5) \( j(x) = \sqrt{x} \): increasing on \( [0, +\infty) \); local minimum at \( (0,0) \).

6) \( k(x) = \sqrt[3]{x} \): increasing on \( (-\infty, +\infty) \); no local minimum or maximum.

7) \( l(x) = e^x \): increasing on \( (-\infty, +\infty) \); no local minimum or maximum.

8) \( m(x) = \ln(x) \): increasing on \( (0, +\infty) \); no local minimum or maximum.

Tutorial (4) – Comparing Rates of Change

1) As \( x \) increases, \( l(x) = e^x \) increases faster than \( g(x) = x^2 \).

2) As \( x \) increases, \( f(x) = x \) increases faster than \( m(x) = \ln(x) \).

Tutorial (5) – Even and Odd Basic Functions

1) Even functions: \( g(x) = x^2 \) and \( i(x) = |x| \), since \[ g(-x) = (-x)^2 = x^2 = g(x), \quad i(-x) = |-x| = |x| = i(x). \] Their graphs are symmetric about the \(y\)-axis.

2) Odd functions: \( f(x) = x \), \( h(x) = x^3 \), and \( k(x) = \sqrt[3]{x} \), since \[ f(-x) = -f(x), \quad h(-x) = -h(x), \quad k(-x) = -k(x). \] Their graphs are symmetric about the origin.

3) The functions \( j(x) = \sqrt{x} \), \( l(x) = e^x \), and \( m(x) = \ln(x) \) are neither even nor odd.