- Graph an absolute value function.
- Solve an absolute value equation.
- Solve an absolute value inequality.
Understanding absolute value functions is helpful for calculus because of the unique sharp corner.
Function Continuity Analysis
Consider the function [latex]f(x) = |x - 2| + 1[/latex]. In calculus, you’ll need to analyze where functions are continuous and where they have breaks or corners.
Converting to Piecewise Form
To understand behavior around the critical point, we write [latex]f(x) = |x - 2| + 1[/latex] as a piecewise function.
Converting Absolute Value to Piecewise
- Find where the expression inside equals zero
- Test values on either side of this point
- Write separate expressions for each piece
- Verify continuity at the boundary point
Rewrite [latex]f(x) = |x - 2| + 1[/latex] as a piecewise function.
- For [latex]x - 2 \geq 0[/latex] we get a domain of [latex]x \geq 2[/latex].The positive piece of the absolute value:
[latex]\begin{align} f(x) &= (x - 2) + 1 \\ &= x - 1 \end{align}[/latex]
- For [latex]x - 2 < 0[/latex] we get a domain of [latex]x < 2[/latex].
The negative piece of the absolute value:
[latex]\begin{align} f(x) &= -(x - 2) + 1 \\ &= -x + 3 \end{align}[/latex]
Piecewise form: [latex]f(x) = \begin{cases} -x + 3 & \text{if } x < 2 \\ x - 1 & \text{if } x \geq 2 \end{cases}[/latex]