Exercises: Piecewise functions

Given the following piecewise function:

$f(x) = \begin{cases} 2x+1 & \text{if } x < 0 \\ x^2-3 & \text{if } x \ge 0 \end{cases}$

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

Look at the graph of the function $f(x)$ and determine its domain and its range.

Express the modulus function $f(x) = |3x - 6|$ as a piecewise function.

A central parking lot charges a fee of $40$ pesos for the first hour of stay (or fraction thereof). After the first hour, it charges at a rate of $30$ pesos for each additional hour, billing the extra time proportionally to the exact minutes of stay. Write the piecewise function $C(x)$ representing the total cost in terms of the hours $x$ that the car stays in the parking lot.

Find the boundary point of the following function and determine whether it is continuous at that point:

$f(x) = \begin{cases} 3x-2 & \text{if } x < 2 \\ 5-x & \text{if } x \ge 2 \end{cases}$

Match the following description with the structure of a piecewise function: "An oven is turned on and its temperature rises at a constant rate for 10 minutes until it reaches 200ºC. It is kept at that temperature for 40 minutes to bake a pizza, and finally it is turned off, with its temperature dropping to room temperature (20ºC) at a constant rate over 30 minutes."

Write the time intervals that would define the three sections of the temperature function $T(t)$.

Plot the graph of the following piecewise function in your notebook:

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

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

From the graph of the function $g(x)$, determine its analytical expression knowing that it consists of two half-lines.

A cell phone company offers a data plan that works as follows: a monthly fixed fee of $300$ pesos is paid, which includes the first $5$ GB of downloads. If that limit is exceeded, each additional gigabyte (or proportional fraction) is charged at $40$ pesos.

Write the function $C(x)$ representing the monthly cost in terms of the gigabytes $x$ consumed, and calculate how much a customer who consumed $3$ GB and another who consumed $8.5$ GB will pay.

Determine the value of the parameter $k$ so that the following piecewise function is continuous throughout its domain:

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

Express the function $f(x) = |x - 2| + 3x$ as a piecewise function without using modulus signs, and calculate the values of $f(0)$ and $f(5)$.

A commuter train starts from a station and accelerates at a constant rate until it reaches a speed of $20$ m/s in $10$ seconds. It maintains that cruising speed for $30$ seconds and, finally, brakes uniformly until coming to a complete stop at the next station in $5$ seconds. Write the piecewise function $v(t)$ describing the speed of the train in terms of the time elapsed since starting.

From the graph of the function $h(x)$, which is made up of a parabolic arc and a half-line, find its analytical expression.

An electricity bill for a home has a standing charge of $200$ pesos per month. In addition, the first $100$ kWh consumed are charged at $2.40$ pesos/kWh. If $100$ kWh is exceeded, the excess consumption is charged at $3$ pesos/kWh. Write the function $F(x)$ giving the total bill amount in terms of the kWh $x$ consumed.

Given the piecewise function:

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

Plot its graph and determine its local maximums and minimums, if any.

In a fictional country, income tax is calculated as follows: for the first $200\,000$ pesos of income no tax is paid ($0$%). For the amount exceeding $200\,000$ pesos up to $400\,000$ pesos, $10$% is paid. For the amount exceeding $400\,000$ pesos, $20$% is paid.

Write the piecewise function $I(x)$ calculating the tax payable in terms of income $x$.

Determine the values of the parameters $a$ and $b$ so that the function $f(x)$ is continuous at $x=1$ and passes through the point $(4, 9)$.

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

Once $a$ and $b$ are found, check mathematically whether the function is also continuous at $x=3$.

Express the function $f(x) = |x + 3| - |2x - 4|$ as a piecewise function, removing the modulus signs, and simplify the expression for each of the resulting sections.

A theater offers a ticket price of $200$ pesos if up to a maximum of $50$ people attend. To encourage attendance by large groups, for each person exceeding $50$, the ticket price for all attendees is reduced by $2$ pesos.

Write the total revenue function of the theater $I(x)$ in terms of the number of attendees $x$ (as a piecewise function) and determine how many people must attend for the theater to obtain the maximum possible revenue.

Look at the following graph of the function $p(x)$ and determine its complete analytical expression. From it, state its domain, its range and classify any discontinuities it exhibits.

A test rocket takes off vertically. During the first $10$ seconds, its height in meters is modeled by the linear function $h(t) = 5t$. At $10$ seconds the engine turns off and the rocket follows a parabolic free-fall trajectory $h(t) = at^2 + bt + c$, such that the transition is continuous. Knowing that the rocket impacts the ground at $50$ seconds, find the expression of the parabola and determine the maximum height reached by the rocket.

Given the piecewise function:

$f(x) = \begin{cases} 2x & \text{if } 0 \le x \le 2 \\ k & \text{if } 2 < x \le 6 \end{cases}$

Calculate the value of the parameter $k$ so that the total area of the geometric figure enclosed between the graph of the function and the x-axis ($X$-axis) is exactly $24$ square units.

A water tank with a capacity of $6000$ liters is initially empty. A pump is turned on that pumps water at a constant rate of $50$ liters per minute. After $60$ minutes, a second auxiliary pump is turned on, adding an additional flow rate of $30$ liters per minute. Write the piecewise function $V(t)$ representing the volume of water in the tank in terms of time, and calculate how long in total it will take to fill the tank completely.

Find analytically the points of intersection between the piecewise function $f(x) = \begin{cases} x^2 - 4 & \text{if } x < 0 \\ -2x + 4 & \text{if } x \ge 0 \end{cases}$ and the line $y = x + 2$.