Piecewise-defined functions

Determine the domain and range of the piecewise function whose algebraic expression is:

$f(x) = \begin{cases} -1 & \text{if } -3 \le x < 0 \\ 1 & \text{if } 0 \le x \le 3 \end{cases}$

Given the following piecewise function:

$f(x) = \begin{cases} 2x + 5 & \text{if } x < -1 \\ 3 & \text{if } -1 \le x \le 2 \\ -x + 4 & \text{if } x > 2 \end{cases}$

Calculate the value of the function at the points $x = -3$, $x = -1$, $x = 0$, $x = 2$ and $x = 5$.

Find the axis intercepts of the piecewise function:

$f(x) = \begin{cases} x + 3 & \text{if } x < 0 \\ 2x - 4 & \text{if } x \ge 0 \end{cases}$

Draw the graph of the following piecewise function in your exercise book:

$f(x) = \begin{cases} x + 2 & \text{if } x \le 1 \\ 3 & \text{if } x > 1 \end{cases}$

State whether the function is continuous at $x = 1$.

You can do it on paper and send a photo with your answer.

Draw the graph of the function defined by:

$f(x) = \begin{cases} -2 & \text{if } x < -2 \\ x & \text{if } -2 \le x \le 2 \\ 2 & \text{if } x > 2 \end{cases}$

You can do it on paper and send a photo with your answer.

Draw the graph of the following function and classify its points of discontinuity, if any:

$f(x) = \begin{cases} -2x - 4 & \text{if } x < -1 \\ 2 & \text{if } -1 \le x < 2 \\ x - 1 & \text{if } x \ge 2 \end{cases}$

You can do it on paper and send a photo with your answer.

Express the modulus function $f(x) = |x - 4|$ as a piecewise function, removing the modulus bars.

Express the function $f(x) = |2x + 6| - 3$ as a piecewise function and draw its graph. Find the coordinates of its vertex (the point where the slope changes).

A public car park charges the following rates: $2$ euros for the first hour (or part thereof) and $1$ euro for each additional hour thereafter. Write the algebraic expression of the function $C(t)$ determining the cost in euros in terms of the time $t$ in hours, up to a maximum parking time of $3$ hours.

The temperature $T$ (in $^\circ\text{C}$) of an industrial ceramics kiln varies depending on the time $t$ (in hours) since it is switched on. During the first $2$ hours, the temperature rises from $20^\circ\text{C}$ at a constant rate of $400^\circ\text{C}$ per hour. Then, it remains constant for $3$ hours. Finally, it is turned off and the temperature decreases at a constant rate of $200^\circ\text{C}$ per hour until returning to $20^\circ\text{C}$.

Write the algebraic expression of the piecewise function $T(t)$.

The graph summarises the behaviour of the kiln:

Calculate the value of the parameters $a$ and $b$ so that the following function is continuous on the set of all real numbers $\mathbb{R}$:

$f(x) = \begin{cases} ax + b & \text{if } x < -1 \\ x^2 - 2x & \text{if } -1 \le x < 2 \\ \frac{a}{x} + b & \text{if } x \ge 2 \end{cases}$