1. To sketch the function f(x) = (2/3)x - 3, we first need to find two points that we can later join to sketch the line, for example the x- and y-intercepts.
a) The x-intercept occurs when f(x) = 0, so if f(x) = 0, then:
f(x) = (2/3)x - 3
0 = (2/3)x - 3
3 = (2/3)x (Add three to both sides)
3*(3/2) = x (Multiply both sides by 3/2)
9/2 = x
We have now found the x-intercept at (9/2, 0)
b) The y-intercept occurs when x = 0, so:
f(x) = (2/3)x - 3
f(0) = (2/3)*0 - 3
f(0) = -3
Now we know that the y-intercept is at (0, -3)
c) All that's left is to sketch the graph axes and label them, plot the two points, join them together using a ruler and label their coordinates.
2. The Domain is the range of x-values for which the function exists, and the Range is the range of y-values for which the function exists.
Since there haven't been any constraints specified, we can say that both the Domain and Range are (-∞, ∞), since the graph continues forever both along the x- and y-axis.
(Note that this isn't always the case and would change if, for example, the question stipulated that there was a domain of [0, 5] and you had to find the range. Then, you would calculate the value of y at each end of the domain (if x = 0, y = -3 and if x = 5, y = 1/3) - in my example, the range would thus be [-3, 1/3].)
