Domain: All real numbers. Range: f(x)<4 or f(x)≥ 7
Practice makes perfect
To graph the given piecewise function, we should think about the graph of each individual piece of the function. Then we can combine the graphs on one coordinate plane.
y=3x+4
First we will graph y=3x+4 for the domain x≥ 1. To do so, let's make a table of values. Remember to include values of the function for x≥ 1.
x
3x+4
y=3x+4
1
3( 1)+4
7
2
3( 2)+4
10
3
3( 3)+4
13
4
3( 4)+4
16
We will plot these ordered pairs on a coordinate plane and connect them to get the graph of f(x). Since the endpoint is included, this piece should end with a closed circle.
Looking at the graph, we can see that all of the possible y-values are greater than or equal to 7.
y=x+3
Next, we will graph y=x+3 for the domain x<1. As we did with the previous piece, we will make a table of values. Remember to include values of the function for x<1.
x
x+3
y=x+3
-5
-5+3
-2
-3
-3+3
0
-1
-1+3
2
0
0+3
3
Similarly as before, we will plot the ordered pairs on a coordinate plane and connect them to get the graph of f(x) for x<1. Since the endpoint is not included, we will end the piece with an open circle.
From the graph, we can see that all y-values that are less than 4 will be produced by this portion.
Combining the Pieces
Finally, we can combine the pieces onto one coordinate plane.
Looking at the pieces together, we can see that there are no gaps in the possible values of x. We can see that there is a gap between the first and second pieces of the function. Our range will include all possible values of y that are not part of this gap. We can use these facts to write the domain and range of the function.
Domain:& All real numbers
Range:& f(x)<4 and f(x)≥ 7
