Sketch the graph of the function with the given rule. Find the domain and range of the function.

Sketch the graph of the function with the given rule. Find...

Key Concepts

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the independent variable. This concept helps in understanding how one function can exhibit different behaviors in separate parts of its domain, often leading to features such as jumps or changes in slope at the points where the definition switches.

Domain

The domain of a function represents all possible input values for which the function is well-defined. In the context of piecewise functions, identifying the domain involves considering the union of the intervals specified for each piece, ensuring that each part of the function is defined for the inputs it is associated with.

Range

The range of a function is the set of all possible output values. When dealing with piecewise functions, the overall range is determined by evaluating the outputs produced by each segment over its respective domain and then combining these outputs, while accounting for any gaps or overlaps.

Graphing Functions

Graphing functions involves plotting the input versus the output visually, which provides insights into the behavior, continuity, and key features of the function. For piecewise functions, it is important to graph each segment on its respective interval, meticulously marking any open or closed endpoints, and clearly indicate points where the rule changes.

Transcript

00:01 Problem 48 it's required to grab the function f of x here and to find its domain and range we can see that f of x is a piecewise function it has an expression for an interval from minus infinity to 2 and another expression for another interval from 2 to infinity to graph a function we need points on this graph then connect these points together but we have here two different lines then we will make points to connect this line and make another point to connect this line to get a point on a graph we just substitute by any value of x here in the function and get a corresponding value of f then the first step is to make a table between different values of x and the corresponding value of f to substitute in the first expiration we just need values of x smaller than 2 we can get for example minus 0 it's smaller than 2 1 and we can just take a point 1 .999 that's smaller than 2 and for the other interval we just need values starting from 2 to infinity we can substitute by values of x equal 2 3 and 4 and so on let's get f of 0 to get f of 0 we substitute in the first exhibition it's 4 minus 0 it's 4 to get f of 1 we substitute in the first exhibition it's 1 it's 1 it's 4 minus 1 gives 3 to get f of 1 .999 we just substitute by x equals 1 .999 gives 2 .0 0 .001 4 minus 1 .99 9 9 9 we just substitute by x equals 1 .999 gives 2 .000 it's 2 .001 to get f of 2 we substitute in the first in the second explanation now here by x equals 2 it's 2 multiplied by 2 minus 2 gives 2 2 2 2 multiplied by 2 is 4 minus 2 gives 2 and x equals 3 it's 2 multiplied by 3 you have 6 minus 2 is 4 and 2 multiply by 4 it's 2 multiplied by 4 minus 2 6 the second step is to plot these points and connect them you can see that we have both the values of x and plus the values of f of x which is y then when x equal 0 y equals 4 the first point is here second bear is 1 and 3 near 1 and 3 near 1 and 3 then it's 1 .999 and 2, 1 .999 is just here, almost here, and 2 is almost deep.

03:40 But because this point is not with us, we make it a hollow circle like this.

03:49 And then connect these points together.

03:58 This is for the first line.

04:00 The second line has the first pair, 2 and 2...

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