The absolute value function
A quality-control machine can't tell whether a rod is too long or too short: all it cares about is how far it deviates, in absolute value, from the target length. The absolute value function turns any "signed difference" into an always-positive distance, and that changes the shape of its graph in a very specific way.
From absolute value to a piecewise function
Express the function $f(x) = |3x - 6|$ as a piecewise function, clearly stating the intervals corresponding to each algebraic expression.
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…
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.
The vertex: domain, range and intercepts
Determine the domain and range of the function $h(x) = |x| + 5$. Justify your answer based on the properties of the modulus.
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 exact coordinates of the axis intercepts of the function $h(x) = |x - 3| - 5$.
Quadratic and rational functions with absolute value
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…
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…
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.
Equations and inequalities with absolute value
Find analytically the points of intersection between the modulus function $f(x) = |3x - 2|$ and the constant function $g(x) = 7$.
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…
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.