The modulus function

Express the function $f(x) = |3x - 6|$ as a piecewise function, clearly stating the intervals corresponding to each algebraic expression.

Given the following piecewise function:

$f(x) = \begin{cases} x + 4 & \text{if } x \ge -4 \\ -x - 4 & \text{if } x < -4 \end{cases}$

Write its analytical expression compactly using the modulus function.

Simplify the expression for the function $f(x) = 2x + |x - 4|$ by writing it as a piecewise function. Then, calculate the average rate of change on the interval $[2, 6]$.

Determine the domain and range of the function $h(x) = |x| + 5$. Justify your answer based on the properties of the modulus.

Determine the exact coordinates of the axis intercepts of the function $h(x) = |x - 3| - 5$.

Determine the coordinates of the vertex and the axis intercepts of the function $f(x) = -|2x - 6| + 8$. Do the branches of this function open towards positive or negative y-values?

Express the function $f(x) = |x^2 - 5x + 6|$ as a piecewise function. State in which interval or intervals the original parabola $y = x^2 - 5x + 6$ has been reflected across the x-axis.

Express the rational function with a modulus $f(x) = \left| \frac{2x - 4}{x + 1} \right|$ as a piecewise function. Next, determine its domain and the equations of its horizontal and vertical asymptotes.

Given the function $f(x) = |x^2 - 2x + k|$, determine the value of the parameter $k$ knowing that the graph of the function touches the x-axis at a single point. For that value of $k$, express $f(x)$ without using the modulus.

Find analytically the points of intersection between the modulus function $f(x) = |3x - 2|$ and the constant function $g(x) = 7$.

A machine cuts metal rods with a target length of $250\text{ mm}$. The cutting error function is defined as $E(x) = |x - 250|$. If quality control requires that $E(x) \le 1.5$, set up the corresponding inequality and determine the interval of valid rod lengths.

Aoife is solving the equation $|2x - 8| = -4$ on her exam and writes the following reasoning:

“Since the absolute value splits into two options:

1) $2x - 8 = -4 \implies 2x = 4 \implies x = 2$.

2) $-(2x - 8) = -4 \implies -2x + 8 = -4 \implies -2x = -12 \implies x = 6$.

Therefore, the equation has two solutions: $x = 2$ and $x = 6$”.

1. a) Without solving the equation again, check by direct substitution whether the values $x = 2$ and $x = 6$ are really solutions of the original equation $|2x - 8| = -4$. What do you observe?
2. b) Identify and explain the conceptual error that Aoife made before starting to isolate the unknown. What is the correct, reasoned answer regarding the number of solutions of this equation?

Find analytically the points of intersection of the functions $f(x) = |x + 1|$ and $g(x) = |2x - 5|$. Hint: analyse the different cases depending on the sign of the expressions inside the modulus signs.

The altitude error of an inspection drone during a manoeuvre is modelled using the sum of two modulus expressions: $E(t) = |t^2 - 4t + 3| + |t - 2|$, where $t$ is time in seconds. Express the function $E(t)$ without using modulus signs in the time interval $t \in [1, 3]$.

To solve the equation $|x - 2| + |x - 6| = 8$, two students propose different approaches:

• Method 1 (analytical piecewise): Study the sign of the binomials on the real line, break down the expression into three intervals ($(-\infty, 2)$, $[2, 6)$, and $[6, +\infty)$), solve the equation in each branch, and check the validity of the solutions.

• Method 2 (geometric distances): Interpret $|x - a|$ as the “distance between $x$ and $a$” on the real line, considering the fixed distance between the points $2$ and $6$.

1. a) Apply Method 1: express $f(x) = |x - 2| + |x - 6|$ as a piecewise-defined function and obtain the solutions of the equation algebraically.
2. b) Apply Method 2: reason what the sum of distances to $2$ and to $6$ is for any point located within the segment $[2, 6]$, and geometrically deduce the two solutions of the equation when this sum must be $8$.
3. c) Choose and justify the method: If you are now asked to solve the equation $|x - 2| + |x - 6| = 3$, which of the two methods allows you to give an immediately justified answer without performing lengthy operations? Explain what that conclusion is.