A tutorial using an HTML5 app to explore the graphical and analytical properties of some of the most common functions used in mathematics. The properties explored include graphs, domain, range, intervals of increase or decrease, local extrema, and symmetry (even, odd, or neither). Complete step-by-step answers and explanations are integrated directly below each tutorial section.
Functions to be Investigated
- Linear function \( f(x) = x \)
- Squaring function \( g(x) = x^2 \)
- Cubing function \( h(x) = x^3 \)
- Absolute value function \( i(x) = |x| \)
- Square root function \( j(x) = \sqrt{x} \)
- Cube root function \( k(x) = \sqrt[3]{x} \)
- Natural exponential function \( l(x) = e^x \)
- Natural logarithmic function \( m(x) = \ln(x) \)
Interactive Tutorial Using HTML5 App
Tutorial (1) - Domain and Range of Basic Functions
Tasks:
- Click on the button above "Plot" to start.
- Select a function and examine its graph. Write down its equation (e.g., \( f(x) = x^3 \)). Do this for all functions.
- Find the domain of each function by examining its graph and equation.
- Find the range of each function.
View Solutions & Answers for Tutorial (1)
- Linear function \( f(x) = x \)
Domain: \( (-\infty, +\infty) \)
Range: \( (-\infty, +\infty) \) - Squaring function \( g(x) = x^2 \)
Domain: \( (-\infty, +\infty) \)
Range: \( [0, +\infty) \) - Cubing function \( h(x) = x^3 \)
Domain: \( (-\infty, +\infty) \)
Range: \( (-\infty, +\infty) \) - Absolute value function \( i(x) = |x| \)
Domain: \( (-\infty, +\infty) \)
Range: \( [0, +\infty) \) - Square root function \( j(x) = \sqrt{x} \)
Domain: \( [0, +\infty) \)
Range: \( [0, +\infty) \) - Cube root function \( k(x) = \sqrt[3]{x} \)
Domain: \( (-\infty, +\infty) \)
Range: \( (-\infty, +\infty) \) - Natural exponential function \( l(x) = e^x \)
Domain: \( (-\infty, +\infty) \)
Range: \( (0, +\infty) \) - Natural logarithmic function \( m(x) = \ln(x) \)
Domain: \( (0, +\infty) \)
Range: \( (-\infty, +\infty) \)
Tutorial (2) - Comparing Basic Functions
Tasks: Compare the properties and graphs of related functions.
View Solutions & Answers for Tutorial (2)
1. For \( x \ge 0 \), \( |x| = x \). Hence, the graphs of \( f(x) = x \) and \( i(x) = |x| \) are identical for \( x \ge 0 \). For \( x < 0 \), \( |x| = -x > 0 \), which explains why the graph of \( i(x) = |x| \) lies entirely above the \(x\)-axis for \( x < 0 \).
2. The functions \( l(x) = e^x \) and \( m(x) = \ln(x) \) are inverse functions 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 inverse functions of each other. Hence, their graphs are reflections across the line \( y = x \).
4. For \( 0 < x < 1 \): \[ x^3 < x^2 < x < \sqrt{x} < \sqrt[3]{x} \] For \( x > 1 \): \[ x^3 > x^2 > x > \sqrt{x} > \sqrt[3]{x} \]
Tutorial (3) - Intervals of Increase and Decrease and Local Extrema
Tasks: Determine monotonicity and local extrema for each basic function.
View Solutions & Answers for Tutorial (3)
- \( f(x) = x \): Increasing on \( (-\infty, +\infty) \); no local minimum or maximum.
- \( g(x) = x^2 \): Decreasing on \( (-\infty, 0) \), increasing on \( (0, +\infty) \); local minimum at \( (0,0) \).
- \( h(x) = x^3 \): Increasing on \( (-\infty, +\infty) \); no local minimum or maximum.
- \( i(x) = |x| \): Decreasing on \( (-\infty, 0) \), increasing on \( (0, +\infty) \); local minimum at \( (0,0) \).
- \( j(x) = \sqrt{x} \): Increasing on \( [0, +\infty) \); local minimum at \( (0,0) \).
- \( k(x) = \sqrt[3]{x} \): Increasing on \( (-\infty, +\infty) \); no local minimum or maximum.
- \( l(x) = e^x \): Increasing on \( (-\infty, +\infty) \); no local minimum or maximum.
- \( m(x) = \ln(x) \): Increasing on \( (0, +\infty) \); no local minimum or maximum.
Tutorial (4) - Compare the Rate of Change of Basic Functions
Tasks: Analyze growth rates.
View Solutions & Answers for Tutorial (4)
1. As \( x \) increases, \( l(x) = e^x \) increases significantly faster than \( g(x) = x^2 \).
2. As \( x \) increases, \( f(x) = x \) increases faster than \( m(x) = \ln(x) \).
Tutorial (5) - Identify Even and Odd Basic Functions
Tasks: Categorize functions by symmetry.
View Solutions & Answers for Tutorial (5)
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. Neither even nor odd: \( j(x) = \sqrt{x} \), \( l(x) = e^x \), and \( m(x) = \ln(x) \).