Async MATH1220 – 2.4

Absolute Functions and Equations

• When solving absolute function equations and plotting points:

  • Steps are similar to that of quadratic functions.
  • Use standard points ranging from negative two to positive two.

• Example of Absolute Value Function:

  • Function: $f(x) = |x|$
  • Calculate absolute values:
  • $| -2 | = 2$
  • $| -1 | = 1$
  • $| 0 | = 0$
  • $| 1 | = 1$
  • $| 2 | = 2$

• Plotting Points:

  • Graph the coordinates based on computed values:
  • Point for $x = -2$ is (–2, 2)
  • Point for $x = -1$ is (–1, 1)
  • Point for $x = 0$ is (0, 0)
  • Point for $x = 1$ is (1, 1)
  • Point for $x = 2$ is (2, 2)

• Graph Details:

  • The vertex point of the graph is at (0, 0).
  • The axis of symmetry is the line $x = 0$.

• Finding Domain:

  • Questions to consider:
  • Is there a variable in the denominator? No.
  • Is there a variable within a radical? No.
  • Conclusion: Domain is $(- ext{∞}, + ext{∞})$.

• Finding Range:

  • The graph starts at (0, 0) and increases.
  • Range in interval notation: $[0, + ext{∞})$.

Piecewise Functions

• Definition: A piecewise function has different expressions based on the value of $x$.

• Example of Piecewise Function:

  • Function: h(x)= \begin{cases}
    x^2 & \text{if } x < 0\
    x + 1 & \text{if } 0 \leq x < 3\
    -1 & \text{if } x \geq 3
    \end{cases}

• Plotting Piecewise Function:

  • For $x^2$ when $x < 0$:
  • Calculate points:
    • $$x = -2, h