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Chapter 2: Problem 104

What is a piecewise function?

Short Answer

Expert verified
A piecewise function is a function that is defined by several different sub-functions, each of which applies to a different interval on the number line. For example, the function could look like \(f(x) = x^2\) for \(x < 0\) and \(f(x) = 2x\) for \(x \geq 0\).

Step by step solution

01

Definition of Piecewise Function

A piecewise function is a type of function that is defined by multiple sub-functions, each applying to a different interval in the domain of the function. It is called 'piecewise' because it is defined by different pieces or sections of various functions.
02

Understanding Piecewise Function

A piecewise function is often depicted graphically with a number line indicating the distinct intervals that apply for each sub-function. These intervals are usually separated by a single point, at which the function value may be either one of the sub-function values or a completely separate value.
03

Example of Piecewise Function

For example, consider the following piecewise function f(x): \[ f(x) = \begin{cases} x^2, & \text{if } x < 0 \ 2x, & \text{if } x \geq 0 \end{cases}\] This function takes the form of the square function \(x^2\) for values of \(x\) less than 0, and the form of the linear function \(2x\) for values of \(x\) equal to or greater than 0.

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Most popular questions from this chapter

Exercises \(98-100\) will help you prepare for the material covered in the first section of the next chapter. In Exercises \(98-99,\) solve each quadratic equation by the method of your choice. Use the graph of \(f(x)=x^{2}\) to graph \(g(x)=(x+3)^{2}+1\)

The bar graph shows your chances of surviving to various ages once you reach 60 The functions $$ \begin{aligned} f(x) &=-2.9 x+286 \\ \text { and } g(x) &=0.01 x^{2}-4.9 x+370 \end{aligned} $$ model the chance, as a percent, that a 60 -year-old will survive to age \(x .\) Use this information to solve Exercises \(101-102\) a. Find and interpret \(f(90)\) b. Find and interpret \(g(90)\) c. Which function serves as a better model for the chance of surviving to age \(90 ?\)

determine whether each statement makes sense or does not make sense, and explain your reasoning. The graph of \((x-4)+(y+6)=25\) is a circle with radius 5 centered at \((4,-6)\)

determine whether each statement makes sense or does not make sense, and explain your reasoning. A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is \(x^{2}+y^{2}=25\) at the point \((3,-4)\)

In your own words, describe how to find the distance between two points in the rectangular coordinate system.
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