Piecewise-Defined Functions

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From the course by University of California, Irvine

Pre-Calculus: Functions

345 ratings

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University of California, Irvine

Pre-Calculus: Functions

345 ratings

This course covers mathematical topics in college algebra, with an emphasis on functions. The course is designed to help prepare students to enroll for a first semester course in single variable calculus. Upon completing this course, you will be able to: 1. Solve linear and quadratic equations 2. Solve some classes of rational and radical equations 3. Graph polynomial, rational, piece-wise, exponential and logarithmic functions 4. Find integer roots of polynomial equations 5. Solve exponential and logarithm equations 6. Understand the inverse relations between exponential and logarithm equations 7. Compute values of exponential and logarithm expressions using basic properties

From the lesson

Functions. Graphs of Functions. Transformations of Functions.

In this module, we define the notion of "function". We learn to distinguish between functions and non-functions in graphical, symbolic and tabular form. Finally, we will explore graphs of functions and how these graphs behave under linear transformations.

Finding the Domain and Range from a Graph 7:34

Domain and Intercepts 7:18

Difference Quotient 5:05

Piecewise-Defined Functions 6:31

Meet the Instructors

Dr. Sarah Eichhorn
Associate Dean, Distance Learning & Tenured Lecturer
Dr. Rachel Cohen Lehman

[MUSIC] Let's look at piecewise-defined functions.

A piecewise-defined function is a function that is defined in pieces or

according to different rules depending upon the input.

[SOUND] So for example, this would be considered a piecewise-defined function.

So what does this mean? It means that f is defined by different rules according

to what x is. That is, if x does not equal -1, then

f(x) = -3. Otherwise, if x does equal -1, then f(x)

is equal to -4. So, there are different rules that define

f depending upon the input x. So let's compute f(-5), f(-1), and f(2)

and then we'll graph f. Let's start with the f(-5).

[SOUND] Since -5 does not equal -1, we'll be using this first piece here.

That is f(-5) = -3. Alright,

and what about f(-1)? [SOUND] We'll be using the second piece down here,

because the input is -1, which means f = -4.

And finally, what about f(2)? [SOUND] Well, since 2 is not equal to -1, we'll

be using this first piece. That is f(2) = -3.

So, it remains for us now to graph f. So, let's say that this is the y-axis and

this is the x-axis, and let's say this is -1, and this is -3, and this is -4.

As long as x does not equal -1, f(x) or y = -3 and y = -3 is the equation of this

horizontal line here. However, when x = -1, we have an open

circle, because when x = -1, the y value is -4.

So this would be the graph of f. Let's look at another example.

[SOUND] Let g be defined by this piecewise function.

We're going to find g(-5), g(-2), and g(2) and then we'll graph g.

So let's start with computing g(-5). [SOUND] Now, which of these three

intervals does -5 lie in? It lies in this first interval, doesn't it? Because -5 <

-3, therefore, we're going to use the first

rule or the first piece to compute this. So g(-5) = -4.

Alright, what about g(-2)? Well, which piece or rule are we going to use to

compute this? [SOUND] Well, -2 lies in the second interval, doesn't it? So we'll

use the second rule. That is, g(-2) = -2 -1 which is equal to

-3. And finally, what about g(2)?

[SOUND] Which of the three intervals does 2 lie in? And we have to be careful here,

because we see a 2 here and we see a 2 here.

However, the condition of equality is down in this third interval here.

Therefore, we use this third piece. That is, g(2) = 3.

Okay. So it still remains to graph g. So let's say this is the y-axis and this

is the x-axis, and let's say this is -3, and this is +2, and this is +3, and -4.

When x is strictly less than -3, then y or g(x) = -4.

Therefore, any x less than -3, y = -4. And then, for any x between -3 and 2, the

function is defined to be x - 1, which is the equation of the line with y

intercept equal to -1 and slope 1. So it will look like this.

And right when we get to two, we have an open circle, because looking back over

here, this is a strict inequality. But also looking back over here, we have

the condition of equality at -3, which means, over here we have to close

this circle up. And also, this will be negative one

because the y intercept is negative one and this would be one because the slope

is one. And, then, for x > 2,

g(x) = 3. So, we have a closed circle at 2,

and then, y = 3 is the equation of a straight line.

And this is how we work with piecewise-defined functions.

Thank you and we'll see you next time.

[SOUND]

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