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 $x = 3$.

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

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

A downtown parking lot charges a rate of $\$2$ for the first hour of stay (or fraction thereof). After the first hour, it charges at a rate of $\$1.50$ for each additional hour, billing the extra time proportionally to the exact minutes of the stay. Write the piecewise function $C(x)$ that represents the total cost as a function of the hours $x$ the car remains in the parking lot.

Find the breakpoint of the following function and determine if 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 200°F. 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 (20°F) 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 following piecewise function in your notebook:

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

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

A cell phone company offers a data plan that works as follows: a fixed monthly fee of $\$15$ is paid, which includes the first $5$ GB of download. If this limit is exceeded, each additional gigabyte (or proportional fraction) is charged at $\$2$.

Write the function $C(x)$ that represents the monthly cost as a function of the gigabytes $x$ consumed, and calculate how much a customer who has consumed $3$ GB and another who has 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 the absolute value 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 $20$ ft/s in $10$ seconds. It maintains this cruising speed for $30$ seconds and, finally, brakes uniformly until it comes to a complete stop at the next station in $5$ seconds. Write the piecewise function $v(t)$ that describes the speed of the train as a function of the time elapsed since starting.

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

The electricity bill of a house has a fixed monthly charge of $\$10$. In addition, the first $100$ kWh consumed are billed at $\$0.12$/kWh. If $100$ kWh are exceeded, the excess consumption is charged at $\$0.15$/kWh. Write the function $F(x)$ that gives the total amount of the bill as a function 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 it graphically and determine its relative maxima and minima, if it has any.

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

Write the piecewise function $I(x)$ that calculates the tax to be paid as a function of the 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 $b$ have been found, check mathematically if the function is also continuous at $x=3$.

Express the function $f(x) = |x + 3| - |2x - 4|$ as a piecewise function, eliminating the absolute values, and simplify the expression of each of the resulting sections.

A theater offers a ticket price of $\$10$ if up to a maximum of $50$ people attend. To encourage the attendance of large groups, for each person exceeding $50$, the ticket price for all attendees is reduced by $\$0.10$.

Write the total revenue function of the theater $I(x)$ as a function 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.

Observe the following graph of the function $p(x)$ and determine its complete analytical expression. From it, indicate its domain, its range and classify the discontinuities it presents.

A test rocket takes off vertically. During the first $10$ seconds, its height in feet is modeled by the linear function $h(t) = 5t$. At $10$ seconds the engine is turned 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 hits 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 $6000$ gallons is initially empty. A pump is turned on that introduces water at a constant rate of $50$ gallons per minute. After $60$ minutes, a second auxiliary pump is turned on that adds an additional flow of $30$ gallons per minute. Write the piecewise function $V(t)$ that represents the volume of water in the tank as a function of time, and calculate how much time 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 $y = x + 2$.