(a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function. (d) Ba

step 1 Okay, so let's work with a piecewise defined function. So let our piecewise defined function be

(a) Find the domain of each function. (b) Locate any...

Key Concepts

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. When determining the range from a graph, one examines the vertical extent of the graph, noting any gaps or limitations in the outputs, which is particularly important for functions with distinct pieces or discontinuities.

Intercepts

Intercepts are the points where a function's graph crosses the axes. The x-intercepts occur where the function's output is zero and the y-intercept occurs where the input x is zero. Identifying intercepts helps in graphing the function and understanding its behavior relative to the coordinate axes.

Graphing Functions

Graphing a function involves plotting points or segments on the coordinate plane that represent the function's behavior over its domain. For piecewise functions, each subfunction is plotted over its designated interval, ensuring that any discontinuities or changes in behavior at the boundaries are accurately represented.

Piecewise Functions

Piecewise functions are defined by different expressions within different intervals of the domain. They allow a function to have multiple rules that are applied to distinct parts of the input values, making them useful for modeling situations that have sudden changes or different behaviors under various conditions.

Domain of a Function

The domain of a function is the complete set of all possible input values (x-values) for which the function is defined. Determining the domain involves identifying any restrictions, such as values that cause division by zero or negative values under a square root, though piecewise functions often have domains that are explicitly stated through their interval conditions.

Transcript

00:01 Okay, so let's work with a piecewise defined function.

00:05 So let our piecewise defined function be f of x will be equal to negative 2x plus 3 if x is less than 1, and 3x minus 2 if x is greater than are equal to 1.

00:23 And let's kind of explore this function a little bit and see what we get out of it.

00:28 So the very first thing, let's talk about its domain.

00:30 So its domain are what xs are we allowed to plug in the function? and basically we're going to look here for our domain.

00:39 So this describes it to us because these are our x values.

00:42 So the top one is telling us we want all the xs that are less than one.

00:46 And the bottom one's telling us we want every x that's bigger than or equal to one.

00:51 So when i see that, it means i'm talking about everything.

00:54 So our domain here would be all reels.

00:58 So we could write out all reels.

00:59 Or we do have a special notation of a double spine r that looks like that.

01:06 That would be your domain.

01:08 So let's next talk about intercepts.

01:10 So we have two types of intercepts.

01:12 We have x intercepts and we have y intercept.

01:14 So let's do our x intercepts first.

01:16 So to find x intercept, it means let y equals zero.

01:23 So let me think about how we could do that so that we wanted to where it crosses our x axis.

01:30 So to do that, we're going to set each one equal to zero and see if it fits the domain it's supposed to.

01:36 So let's look at that top one.

01:38 So if i set the top one equal to zero, negative 2x plus three, and then i solve, i would subtract three from both sides.

01:48 Divide both sides by negative two.

01:50 I get x is three halves.

01:52 So three halves is bigger than one.

01:56 So this one didn't work for this piecewise defined.

02:00 Function.

02:01 So that must mean that we could possibly still have an x intercept from the other one, but definitely not from the top one.

02:08 So let's let y equals zero for the bottom one, subtract, or i'm sorry, add two to both sides, divide by three, and we get x is two -thirds.

02:20 But that's not bigger than one.

02:23 Two -thirds is less than one.

02:24 So that is also not going to be an x -intercept.

02:29 All right, so let's now try.

02:31 So we have no x intercept.

02:32 Again, let's say that in a different way.

02:34 No x intercept means we're never going to cross the x axis.

02:37 When we go to draw this in a minute, we should see it never hits the x axis.

02:42 Let's find our y intercept.

02:44 To find a y intercept, we let x equal zero.

02:47 Well, if we let x equals zero, looking at the two pieces of our function, x equals zero is a number that's less than one.

02:53 So i would want to use this top formula...

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