To graph the given piecewise function f(x), we should think about the graph of each individual piece of the function. Then we can combine the graphs on one coordinate plane.
y= 12x-1
First we will graph y= 12x-1 for the domain x > 3. To do so, let's make a table of values. Remember to include values of the function for x>3.
x
1/2x-1
y=1/2x-1
4
1/2( 4)-1
1
6
1/2( 6)-1
2
8
1/2( 8)-1
3
We will plot these ordered pairs on a coordinate plane and connect them to get the graph of f(x). Since the endpoint is not included, we will end the piece with an open circle.
Looking at the graph, we can see that all of the possible y-values are greater than 0.5.
y=- 2x+3
Next, we will graph y=- 2x+3 for the domain x≤ 3. As we did with the previous piece, we will make a table of values. Remember to include values of the function for x≤ 3.
x
- 2x+3
y=-2x+3
0
-2( 0)+3
3
1
-2( 1)+3
1
2
-2( 2)+3
-1
3
-2( 3)+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≤ 3. Since the endpoint is included, this piece will end with a closed circle.
From the graph, we can see that all y-values that are greater than or equal to -3 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 also see there are no gaps in the possible values of y but that they are all greater than or equal to -3. We can use these facts to write the domain and range of the function.
Domain:& All real numbers
Range:& f(x)≥-3
