State the domain and range of the step function whose graph is shown.
First let’s recall what is meant by the terms domain and range. The domain of a function is the complete set of values on which the function acts. You can think of them like the input values for the function. When a function is represented on a graph of 𝑥 against 𝑦, the domain of the function is the 𝑥 values for which the graph is drawn.
Looking at the graph more closely, we can see that this step function has been drawn for 𝑥 values from negative eight all the way up to 6.5. The open circle at negative eight indicates that this end of the interval is open, whereas the closed circle at 6.5 indicates that this end of the interval is closed.
We can express the domain as an inequality: 𝑥 is between negative eight and 6.5 with a strict inequality at the lower end and a weak inequality at the upper end. Now let’s recall what is meant by the range of a function? The range is the complete set of values that the function takes. You can think of these as the output values of the function.
On a graph of 𝑥 against 𝑦, this will be all of the 𝑦 values for which the function has been plotted. As this function is a step function, its range isn’t an interval but rather a finite set of values. For 𝑥 values between negative eight and negative two, the function takes the value of four.
For 𝑥 values between negative two and zero inclusive, the function takes the value negative two. And for 𝑥 values from zero up to and including 6.5, the function takes the value 0.5. Writing these values in ascending order, we can see that the range of the step function is a set of three values: negative two, 0.5, and four.
So the domain of the step function is the interval negative eight is less than 𝑥 is less than or equal to 6.5, and the range of the function is the set of values negative two, 0.5, four.