Let's write rules for the functions, a(w) and p(w), first and then we can compare their slopes and y-intercepts. Finally, we will determine their domains and ranges and compare those as well.
Rule for a(w)
When buying apples at a farmer's market, we pay $1.10 for each pound. Therefore, when buying w pounds, we will pay
1.10w dollars.
Thus, the function a(w), which represents the cost of buying w pounds of apples, can be written as:
a(w)=1.10w.
Rule for p(w)
We must determine the rule for p(w) by looking at the graph. Let's identify the coordinates of two points on the graph to find the slope first!
Now we can substitute (0,0) and (10,14) into the Slope Formula!
m = y_2 - y_1/x_2 - x_1
m = 14 - 0/10 - 0
m = 14/10
m = 1.40
The slope of this function is 1.40. Also, because the graph passes through the origin, we know that the y-intercept is 0. Therefore, we can write the final rule for p(w).
p(w)=1.40w
Comparison of slopes and y-intercepts
We now have the rules for the two functions, a(w) and p(w).
a(w)&= 1.10w
p(w)&= 1.40w
We can see that the slope of p(w) is greater than the slope of a(w). However, both functions have the same y-intercept, y=0.
Domains
A domain is the set of all possible input values for a function. We know that apples can be bought at a farmer's market up to 10 pounds at a time. This means that we can buy the maximum of 10 pounds and a minimum of 0 pounds, if do not buy any apples. Then, the domain of a(w) is:
0 ≤ w ≤ 10.
From the graph, we can see that the possible values of w for p(w) also start from w=0 and end at w=10. However, we do not know what happens after that, so we can assume that the graph continues on forever. You can buy as many pears as your heart desires! Thus, the domain of p(w) is:
w ≥ 0.
Ranges
Since the domain of a(w) is 0≤ w ≤ 10, its range is the set of all real values from a(0) to a(10). Luckily, we know a(0) is the y-intercept of a(w). From the function rule, and knowing about slope-intercept form, we can tell that it is 0. Let's find a(10) by substituting w=10 in a(w).
a(w)=1.10w
a( 10)=1.10( 10)
a(10)=11
The range of a(w) is:
0 ≤ a(w) ≤ 11.
We can identify the range of p(w) by looking at the outputs of p(w) on the graph. The function starts at p(w)=0 and, since we have assumed that it continues on forever, it does not have a maximum value. Therefore, its range is:
p(w) ≥ 0.
