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.

Graph of a piecewise function with segments from (-1,-1) to (0,0) and from (2,2) extending rightward.

Accepted Answer

Hello everyone. For the following graph, we have been asked to write its equivalent piecewise function and write its domain and range. Our graph has a section in quadrant two that decreases from left to right, passing through the points negative 3 12, negative 28 ending at the 0.0. We also have a section that begins at the 0.0, negative five but it, the 50.0 negative five is not included and continues as a horizontal line at negative five. We have four answer choices but we are gonna write our function first and then look at them. So first we're going to write our piece wise function. I'm gonna start with the section that's in quadrant two. I want to find what the equation would be for that line. So I'm gonna start by finding the slope of the points negative 3, 12, a negative 28. Recalling that slope is the change in Y divided by the change in X. I'm gonna do 12 minus eight over negative three minus negative two. And this will give me a slope of four over negative one which is the same as negative four from there. I'm going to find what the equation is using that slope in the point negative 3 12. So if I plug that into Y equals M X plus B form where negative four is my M and negative three is my X 12 is my Y. We're gonna find B so we're gonna find what the Y intercept is. So 12 equals negative four multiplied by negative three and plus B. So 12 equals negative four, multiplied by negative three is positive 12 plus B subtracting 12 zero equals B. So this means that the equation at the point negative 3, 12 is Y equals negative four X. So our Y intercept is zero, which means the entire section where X is less than or equal to zero is represented by the function Y equals negative four X. So this is when X is less than or equal to zero. So this is the first part of our piecewise function. Next, I'm looking at the section that is in the fourth quadrant, it is a horizontal line that begins at the 0.0, negative five horizontal lines are set up as Y equals and they have no slope. The slope is zero. So here we can see that Y equals negative five. And we also have a circle, an open circle on the 0.0, negative five. So that's not included in our solution, which means for all X values greater than zero, the function is Y equals negative five. So what this would look like if I were to write it out as a piecewise function is F of X equals open bracket negative four X if X is less than, or equal to zero and negative five if X is greater than zero. So this is part of our answer and we will go match it to our answer choices in a moment. But first, we're gonna find the domain in range. Recall that domain is the X values that are covered by the graph. So looking at my graph, is there any X value that is not represented here? Well, the only hole we see in the graph is at zero negative five and the X value of zero is covered in the other half of the graph. This means our domain is all real numbers. There is not an X value you could pick that would not be represented on the graph. So that's the next part of our solution for the range. Recall that range represents the Y values that are on the graph. So does it go on forever both up and down? Well, looking at the first section of the graph that we found before the one that starts at 00 from there, it increases forever. It looks like it does not stop. So it, our range definitely includes the 0.0. So square bracket and goes to positive infinity. Now, does it go below that? Does it represent any other values there? In the bottom half of the graph, the only Y value that is covered is at negative five. So our range, yes, curly bracket, negative five union with zero to positive infinity including zero. This is our solution. And now we need to find which answer choice does that match to? Here are four answer choices. Each of them is similar but slightly different. So again, what we had as our answers were that the piecewise function is F of X equals open curly bracket, negative four X, if X is greater than oh, I'm sorry, less than or equal to zero and negative five FX is greater than zero. We also said that our domain is all real numbers. So from negative infinity to positive infinity and our range is open curly bracket, negative five close curly bracket union with zero to positive infinity with the zero included. So looking at our choices and to see which one that matches with. So far, just looking at the piecewise function, it has to be B or D checking the domain and range uh for B that does not work out. So our answer choice is answer choice D that is the one that matches both our piece wise function and our domain and range. Have a nice day.

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

Key Concepts

Piecewise-Defined Functions

Domain and Range

Interpreting Graphs and Coordinates

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