Absolute value functions (HL)

When evaluated over a value, it returns the (non-negative) distance from zero.

The analogous modulus is defined for a complex number.

Contents

• Properties
• Composition of absolute value and functions
  • $\bm{f(\lvert x\rvert)}$
  • $\bm{\lvert f(x)\rvert}$
• Solving absolute value equations and inequalities
  • $\bm{\lvert ax + b\rvert \geq k\lvert cx + d\rvert}$
  • $\bm{\lvert f(x)\rvert \geq k}$
  • Strategy to solve absolute value equations
  • Strategy to solve absolute value inequalities

Properties

Composition of absolute value and functions

This extends ideas from composite functions to involve the absolute value function.

$\bm{f(\lvert x\rvert)}$

This composition throws away the left half of $f(x)$ where $x < 0$, and creates an even function using the $x \geq 0$ portion.

Unless originally $f'(0) = 0$, this new function is not differentiable at $x = 0$.

Also the new function has the same range as $f(x), x \geq 0$.

Conceptually, $f(\lvert x\rvert)$ converts dependence on position into dependence on distance.

$\bm{\lvert f(x)\rvert}$

This composition reflects every portion of $f(x) < 0$ across the $x$-axis to make the entire range non-negative.

If $f$ is an odd or even function, then $\lvert f(x)\rvert$ is an even function.

This typically creates non-differentiable points at each zero or $x$-intercept. Be care when doing calculus by hand on $\lvert f(x)\rvert$.

The distance traveled while following some one-dimensional velocity function $v(t)$ from time $t_1$ to $t_2$ is the unsigned area under the absolute value of the velocity function.

Solving absolute value equations and inequalities

$\bm{\lvert ax + b\rvert \geq k\lvert cx + d\rvert}$

As surprising as it may sound, many absolute value linear inequalities can involve quadratics.

If it is instead finding $\lvert ax + b \rvert < k$ then instead you need the single interval between the two roots of the equality.

One way to check the solutions is to substitute in different values, as you would after solving an equation or inequality.

$\bm{\lvert f(x)\rvert \geq k}$

If asking to solve $\lvert f(x)\rvert \geq k, \, k \geq 0$, then solutions satisfy

If asking to solve $\lvert f(x)\rvert \leq k, \, k \geq 0$, then solutions satisfy

Strategy to solve absolute value equations

Get the absolute value(s) on one side, and square both sides, to turn an linear equation into a quadratic. In some cases, you may need to square both sides twice, while doing some simplification in between and afterwards.

Check all solutions at the end to verify it solves the original equation.

Strategy to solve absolute value inequalities

First solve the equality.

For each interval separated by the roots of the equality, test a value and see if it satisfies the original equation.