Exercises: Modulus function

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

Calculate the numerical value of the function $g(x) = |-x^2 + 4x|$ for the points $x = -1$, $x = 0$, $x = 2$ and $x = 5$.

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

Given the following graph of a modulus function, deduce its analytical expression in the form $y = |x - a|$.

Solve the equation $|2x - 8| = 0$ and explain what this result represents geometrically with respect to the graph of the function $f(x) = |2x - 8|$.

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.

Express the quadratic function with absolute value $f(x) = |x^2 - 25|$ as a piecewise function.

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

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?

An absolute value function of the form $f(x) = |x - a| + b$ has the following characteristics in the Cartesian plane:

• Its graph is “V”-shaped with its vertex at the point $(4, -3)$.

• It intersects the x-axis (axis of abscissas) at two points that are symmetric with respect to the vertical line through the vertex.

1. a) Reasonably determine the values of the parameters $a$ and $b$.
2. b) Analytically find the two x-intercepts by solving the equation $f(x) = 0$.
3. c) Rewrite $f(x)$ as a piecewise-defined function and reason whether the found vertex corresponds to an absolute minimum or an absolute maximum of the function.

In a laboratory, the temperature deviation of a chemical reaction from the ideal temperature of $22^\circ\text{C}$ is given by the function $D(t) = |2t - 12|$, where $t$ is the time in minutes. Express $D(t)$ as a piecewise function and determine at what instant the deviation is zero.

A company's profit margin, in thousands of euros, relative to its annual target is modelled by the function $M(x) = |x^2 - 40x + 300|$, where $x$ is the number of months elapsed. Express the function as a piecewise function to identify in which periods the company was exactly on target, below it (before applying the modulus), or above it.

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 $[2, 6]$.

In a billiards simulation, a ball bounces off the cushion located on the x-axis. The path of the ball follows the function $y = 4 - |x - 3|$, where $x$ and $y$ are measured in units of the simulator's grid.

Determine analytically (without drawing the graph, or using it only to check):

1. a) The coordinates of the point where the ball reaches its maximum distance from the cushion.

1. b) The coordinates of the points where the ball would hit the X-axis.

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.

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.

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?

In a mountain stage of the Rás Tailteann cycle race passing through County Kerry, the elevation profile of a $20\text{ km}$ section is modelled by the function:

$h(x) = 60 \times |x - 8| + 450$

where $x$ is the distance covered in kilometres from the start of the section ($0 \le x \le 20$) and $h(x)$ is the altitude above sea level in metres.

1. a) Identify the coordinates of the vertex of the graph of $h(x)$ and interpret its physical meaning in the race (kilometre mark and altitude of the valley floor).
2. b) Express the function $h(x)$ as a piecewise-defined function for the domain interval $[0, 20]$.
3. c) The organizers set up a liquid refreshment station at all points along the section where the altitude does not exceed $690\text{ m}$. Set up an absolute value inequality for this condition and calculate the kilometre range (interval of $x$) where this refreshment zone will be located.

Consider the function $f(x) = |x^2 - 6x + 5|$ and the horizontal line $y = 3$.

1. a) Estimate with reasoning:

• Find the roots and the coordinates of the vertex of the inner parabola $g(x) = x^2 - 6x + 5$.

• Explain how the absolute value acts on the part of the parabola located below the x-axis and deduce the maximum height reached by the reflected “crest”.

• From the height of that crest and the general behaviour of the graph, reasonably estimate how many real solutions the equation $|x^2 - 6x + 5| = 3$ will have.

1. b) Check algebraically:

• Solve the equation $|x^2 - 6x + 5| = 3$ by setting up the two equations that arise when removing the absolute value.

• Find all exact solutions and confirm whether your previous estimate regarding the number of solutions was correct.

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.

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.

The following graph shows a function with a nested modulus of the form $f(x) = ||x| - c|$. Observe the vertices and intercepts, deduce the value of $c$ and write the expression for the piecewise function on the interval $x \in [-4, 4]$.

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 $t \in [1, 3]$.

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.

In signal processing, an inverted triangular wave can be modelled using the function $S(t) = A - |B t - C|$. Determine the parameters $A, B, C > 0$ knowing that the signal reaches its maximum value of $12\text{ V}$ at time $t = 5\text{ s}$, and crosses the x-axis (value $0\text{ V}$) at times $t = 2\text{ s}$ and $t = 8\text{ s}$.

The graph shown corresponds to the function $f(x) = |x^2 + bx + c|$. From the observable key points (roots and local maximum), set up a system of equations and determine the values of the parameters $b$ and $c$.

Find the domain of definition of the radical function $h(x) = \sqrt{8 - |x^2 - 8|}$. To do so, set up and solve the corresponding inequality involving modulus.

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.

Consider the family of quadratic functions with absolute value given by $f_k(x) = |x^2 - 4x + k|$, where $k \in \mathbb{R}$ is a constant.

1. a) Reasonably determine for which values of the parameter $k$ the graph of $f_k(x)$ undergoes no reflection across the x-axis (that is, it coincides across its entire domain with the original parabola $y = x^2 - 4x + k$).
2. b) Justify, by studying the roots of the inner polynomial, how many sharp points (corners on the x-axis where the function abruptly changes direction) the graph of $f_k(x)$ has in each of these three cases: $k > 4$, $k = 4$, and $k < 4$.
3. c) Inverse problem: Knowing that the horizontal line $y = 5$ intersects the graph of $f_k(x)$ at exactly three distinct points, deduce with justification the value of the parameter $k$ and calculate the exact coordinates of those three intersection points.

Topic challenge. Research the electricity demand on the EirGrid Smart Grid Dashboard website for a full day. Your final product will be a brief analytical report prepared using GeoGebra or a spreadsheet.

1. Data and derivative: Download the $24$-hour hourly data $D(t)$. Model the curve and identify its domain, range, and monotonicity. Determine and interpret graphically the consumption peaks and valleys by solving $D'(t) = 0$.

2. Composition, inverse, and transformations: If the average hourly price is $C(x) = 0.15x$ €/MWh, find with reasoning the composite function $(C \circ D)(t)$. Select a strictly increasing morning section to deduce and interpret the inverse function $D^{-1}$. Then apply a translation $D(t - 2)$ to simulate a delay in consumption habits.

3. Parity and absolute value: Center the time axis at $12.0$ h and formally check whether the curve exhibits parity (even or odd symmetry). Design the deviation function $F(t) = |D(t) - D_{\text{average}}|$, analyzing the effect of the absolute value on its critical points.

The following will be assessed:

• Rigor in the characterization of $D(t)$ and in the graphical interpretation of the derivative function.

• Precision in the calculation and meaning of the composite function, the local inverse, and the translation.

• Appropriate mathematical justification of parity and the absolute value operator.

• Expository clarity and graphical correctness in the evidence provided with GeoGebra.