Give a rule for each piecewise-defined function. Also give the...

Question

Give a rule for each piecewise-defined function. Also give the domain and range.

Accepted Answer

Hello everyone. For the following graph, we are going to write an equivalent piecewise function and write its domain and range. The graph we're given is a horizontal line beginning at the 0.5 without that point included continuing in the positive X value direction. The other section of the graph begins at the 0.0, negative three is also a horizontal line and continues in the negative X value direction. We have four answer choices. Only one is visible. Currently on the screen, we'll scroll down to the others after we write our function and find the domain and range beginning with our piecewise function. I'm gonna first look at the section where the graph starts at the 0.5 and continues in the positive X direction. It's a horizontal line and I recall that horizontal lines have a slope of zero. So horizontal lines are set up as a of X equals the Y value of the coordinate. So in this example, we have the 0.5. So F of X is gonna equal five here. This occurs when the X value is greater than zero. It's not equal to zero as our 00.5 is an open circle, not a closed circle. The next section is again a horizontal line. This one begins at the 10.0 negative three. So our function would be F of X equals negative three. And this is gonna happen when X is less than, or equal to zero. And I know this one's or equal to because the 00.0 negative three is a closed circle. So writing this out as a piecewise function I have F of X equals open curly bracket five. If X is greater than zero, negative three, if X is less than or equal to zero. So this is the first part of our solution. We have our piecewise function written next, we need to find our domain and range. Recall that domain is the X values that a graph covers. So looking at this horizontally thinking from left to right, is there any value that is not covered by a graph for an X value? The only spot that might have been questionable is when X equals zero because we have an open circle at the 00.5. But since that's a closed circle on zero, negative three, all the X values are represented, which means our domain is all real numbers. So it goes from negative infinity to positive infinity. Two thirds of the problem down one third to go. So we need to find the range of this graph recall range is the Y values that are covered by the graph. So in this case, our range is gonna be very small because we have two horizontal lines. The Y values represented here on this graph are negative three because that's where the horizontal line is that goes with zero, negative three in five. Because again, that's our horizontal line for 05, no other Y values are covered at all. So we don't have to list them. These are in curly brackets to show that they are not an interval but two separate points. Now that we know our answers, let's look at all of our answer choices and decide which one fits best. Now that all four of our answer choices are in view, I'm going to again rewrite what we had as our answer, which is F of X equals open curly bracket five. If X is greater than zero, negative three, if X is less than or equal to zero, we had a domain of all real numbers. So negative infinity to positive infinity and a range that was the points. So in curly brackets negative 35, comparing this to our answer choices. It looks like our response is answer choice. A, they wrote their piecewise function the opposite way I did, but they still have the same functions. So it is answer choice. A have a nice day.

Give a rule for each piecewise-defined function. Also give the domain and range.

Key Concepts

Piecewise-Defined Functions

Domain of a Function

Range of a Function

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