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 $h(x) = |x - 3| - 5$.

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 pesos, relative 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 as a piecewise function to identify in which periods the company was exactly on target, below it (before applying the absolute value), 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?

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.