Exercises: Modulus function

Express the function $f(x) = |3x - 6|$ as a piecewise function, clearly indicating 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 the 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 in a compact form using the modulus.

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

Determine the exact coordinates of the points of intersection with the Cartesian axes of the function $h(x) = |x - 3| - 5$.

In a laboratory, the deviation of the temperature 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 dollars, with respect to its annual target is modeled by the function $M(x) = |x^2 - 40x + 300|$, where $x$ is the number of months elapsed. Express the function piecewise to identify in which periods the company was exactly on target, below it (before applying the modulus) or above it.

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

In a billiards simulation, a ball bounces against the cushion located on the x-axis. The trajectory of the ball follows the function $y = 4 - |x - 3|$. Observe the representation and analytically determine the coordinates of the point where the ball reaches its maximum separation from the cushion and the points where it would impact 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 lengths for the rods.

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 in the x-axis.

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

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

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

The altitude error of an inspection drone during a maneuver is modeled by the sum of two moduli: $E(t) = |t^2 - 4t + 3| + |t - 2|$, where $t$ is the time in seconds. Express the function $E(t)$ without using moduli 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: analyze the different cases according to the sign of the expressions inside the modulus.

In signal processing, an inverted triangular wave can be modeled with 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 that it 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 singular 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 irrational function $h(x) = \sqrt{8 - |x^2 - 8|}$. To do this, set up and solve the corresponding modulus inequality.