Understanding Absolute Value and Piecewise Functions
1. Absolute Value Functions
The absolute value function is defined as: \[ f(x) = |x| = \begin{cases} x, & \text{if } x \geq 0, \\ -x, & \text{if } x < 0. \end{cases} \] It outputs the magnitude of a number, ignoring its sign.
Domain and Range of Absolute Value Functions
Domain: The domain of an absolute value function is all real numbers: \[ (-\infty, \infty). \] Range: Since the absolute value of any number is non-negative, the range is: \[ [0, \infty). \]
Graphing Absolute Value Functions
The graph of \(f(x) = |x|\) is a "V"-shaped curve with a vertex at the origin \((0, 0)\). It increases for \(x > 0\) and decreases for \(x < 0\), reflecting the definition of the function.
Example 1: Simple Absolute Value Function
Consider: \[ f(x) = |x|. \] Domain: \[ (-\infty, \infty). \] Range: \[ [0, \infty). \]
Example 2: Absolute Value with a Transformation
Consider: \[ f(x) = |x - 2| + 3. \]
- The transformation \(x - 2\) shifts the graph 2 units to the right.
- The transformation \(+3\) shifts the graph 3 units up.
Domain: : \[ (-\infty, \infty).\]
Range: Since the minimum value of \(f(x)\) occurs at the vertex \((2, 3)\), the range is: \[ [3, \infty). \]
Word Problem: Absolute Value
Problem: A drone is flying at a height of 50 m above sea level. If it descends to a height of 20 m and then climbs to 80 m, what is the total distance traveled by the drone? Use the absolute value function.
Solution:
Let the height of the drone be represented as \(h(t)\). The total distance traveled is:
\[
\text{Distance} = |20 - 50| + |80 - 20|.
\]
Calculate step-by-step:
- \(|20 - 50| = 30\).
- \(|80 - 20| = 60\).
2. Piecewise Functions
A piecewise function is defined using different expressions for different parts of the domain. For example: \[ f(x) = \begin{cases} x^2, & \text{if } x \leq 0, \\ 2x + 1, & \text{if } x > 0. \end{cases} \] Here, \(f(x)\) uses \(x^2\) for \(x \leq 0\) and \(2x + 1\) for \(x > 0\).
Domain and Range of Piecewise Functions
Domain: The domain of a piecewise function is the union of the domains of all its parts.
Range: The range is the set of all output values the function can take, which may need to be evaluated for each piece.
Example 3: Simple Piecewise Function
Consider:
\[
f(x) =
\begin{cases}
x + 2, & \text{if } x \leq -1, \\
x^2, & \text{if } x > -1.
\end{cases}
\]
Domain:
\[
(-\infty, \infty).
\]
Range:
- For \(x + 2\) when \(x \leq -1\), the range is \((-\infty, 1]\).
- For \(x^2\) when \(x > -1\), the range is \([0, \infty)\).
Combining these:
\[
(-\infty, \infty).
\]
Example 4: Piecewise Function with Absolute Value
Consider:
\[
f(x) =
\begin{cases}
|x|, & \text{if } x \leq 2, \\
3x - 4, & \text{if } x > 2.
\end{cases}
\]
Domain:
\[
(-\infty, \infty).
\]
Range:
- For \(|x|\) when \(x \leq 2\), the range is \([0, 2]\).
- For \(3x - 4\) when \(x > 2\), the range is \((2, \infty)\).
Combining these:
\[
[0, \infty).
\]
Word Problem: Piecewise Function
Problem: A taxi company charges $3 for the first mile and $2 for every subsequent mile. Write a piecewise function to represent the cost \(C(x)\) of a ride for \(x\) miles.
Solution:
The cost function is:
\[
C(x) =
\begin{cases}
3, & \text{if } 0 < x \leq 1, \\
3 + 2(x - 1), & \text{if } x > 1.
\end{cases}
\]
Simplify:
\[
C(x) =
\begin{cases}
3, & \text{if } 0 < x \leq 1, \\
2x + 1, & \text{if } x > 1.
\end{cases}
\]
3. Practice Questions
- Find the domain and range of \(f(x) = |x + 3| - 2\).
- Graph the piecewise function: \[ f(x) = \begin{cases} x^2, & \text{if } x \leq 1, \\ x + 2, & \text{if } x > 1. \end{cases} \]
- Solve for \(x\): \(|x - 5| = 7\).
- Find the total distance traveled by a car if it moves 10 km north, 15 km south, and 20 km east. Use absolute value.
- Write a piecewise function for: \[ f(x) = \begin{cases} x^2 - 4, & \text{if } x < 0, \\ 2x + 1, & \text{if } x \geq 0. \end{cases} \]
- Find the range of \(f(x) = |2x - 3| + 1\).
- Determine if the piecewise function is continuous: \[ g(x) = \begin{cases} 3x + 1, & \text{if } x < 2, \\ 7, & \text{if } x = 2
Add comment
Comments