Graph each function. Identify the domain and range. f(x)={ -x if x ≤ 3 2 if x>3 .

Name: Problem 26, Chapter 2: Algebra 2
Uploaded: 2020-03-04T19:30:14-08:00
Duration: 3 min 39 s
Channel: Kami Dupree
Description: Graph each function. Identify the domain and range. f(x)={ -x if x ≤ 3 2 if x>3 .

Graph each function. Identify the domain and range. f(x)={ ...

Key Concepts

Piecewise Functions

Piecewise functions are functions defined by multiple sub-functions, each of which applies to a specific interval of the domain. They are used to express situations where a function behaves differently over various parts of its domain. Understanding piecewise functions involves analyzing the conditions under which each sub-function is valid and determining how these pieces fit together to form the overall function.

Domain

The domain of a function refers to the complete set of allowable input values (typically x-values) for which the function is defined. When dealing with piecewise functions, the domain is determined by the union of all intervals over which the different sub-functions are specified. Identifying the domain requires careful consideration of any restrictions or conditions imposed on the variable.

Range

The range of a function is the set of all possible output values (typically y-values) that the function can produce. Determining the range involves evaluating each sub-function over its respective interval and combining the results. For piecewise functions, the overall range is the union of the ranges of the individual pieces, taking into account any potential gaps or overlaps.

Graphing Functions

Graphing functions involves plotting the function's output against its input on a coordinate system. This process is especially important for piecewise functions, as each segment must be accurately represented based on its defining formula and the specific portion of the domain it covers. Graphing helps visualize the behavior of the function, clarify discontinuities or jumps between pieces, and assist in understanding the function's overall structure.

Transcript

00:02 Okay, this problem we want to graph the function, the piecewise function, f of x is opposite of x for x less than equal to three, or absolute value, or rather, f of x is two for values of x greater than three.

00:15 Because this is a piecewise function, and i want you to be able to see how each piece factors into the final picture of the graph.

00:22 I'm going to graph the two pieces and talk about them separately and color code them.

00:26 So the first function that we want to graph is f of x equals the opposite of x.

00:33 But that values, that that's only true for x's values that are, x values that are less than are equal to three.

00:39 So in kind of getting an idea of what this might look like, we want to only take values that are three or less.

00:45 And so that might keep, we want to keep that in mind because we don't want to be putting any larger values than three in here.

00:53 And then the function values we're going to get out are just the opposites.

00:56 Right, opposite of x.

00:58 So this is pretty easy to do.

00:59 If x is three, y will be the opposite of three.

01:03 If x is two, y will be the opposite of two.

01:05 If it's one, it's the opposite of one.

01:06 Of course, zero has no opposites.

01:08 So the opposite of zero is zero.

01:10 So therefore, we'll just have zero.

01:12 If x is a negative one, right, we'd have the opposite of negative one.

01:16 So that's going to be positive one.

01:18 Positive one here.

01:19 And if we have the opposite of negative two, we're going to have positive two.

01:22 So if we start to plot those points, when x is three, y is negative one.

01:26 Negative 3.

01:28 X is 2.

01:28 Y is negative 2.

01:29 X is 1 .y is negative 1 .0.

01:32 X is negative 1.

01:33 Y is negative 1.

01:34 X, negative 2, positive 2.

01:36 We can see, we know that this is a line...

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