Question

Find other examples of real-world situations that can be modeled by piecewise defined functions, and express the models in function notation.

Short answer

Example: A boat ride charges $15 per person for a round trip journey with a group of 1 to 9 people or a fixed $100 fee for a group of 10 or more people. We can write a function relating the number of people, $x$, to the total charge,$C\left(x\right)$

Piecewise function is:

$C\left(x\right)=\left\{\begin{array}{c}15x,if0<x<10\\ 100,ifx>10\end{array}\right\}$

Step-by-step solution

Step 1. Use the definition of functions.

A function is defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x, y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

Step 2. Analyze the information given.

Examples of real-world situations that can be modeled by piecewise defined functions.

Step 3. Construct an example function.

Suppose our hypothetical function is defined such as:

A boat ride charges $15 per person for a round trip journey with a group of 1 to 9 people or a fixed $100 fee for a group of 10 or more people. We can write a function relating the number of people, $x$, to the total charge,$C\left(x\right)$

Two different cases are possible.

Case-1:

For x-values under 10, $C\left(x\right)=15x$.

Case-2:

For values of $x$that are 10 or greater, localid="1646240035387" $C\left(x\right)=100$

Step 4. define the function algebraically.

Thus piecewise function can be constructed as:

$C\left(x\right)=\left\{\begin{array}{c}15x,if0<x<10\\ 100,ifx>10\end{array}\right\}$