Let's write the rules for both functions first. Then we can identify and compare their domains, ranges, slopes, and y-intercepts. Next, we can graph them on one coordinate plane and compare the benefits and drawbacks of each compensation plan. Finally, we will make a decision on which compensation plan Lindsay should take.
Rule for f(x)
We know that plan F offers a $450 base weekly salary. This means that, even if Lindsay makes $0 in sales during one week, she will still earn $450. Therefore, the y-intercept of f(x) is 450. Additionally, she earns a 10 % commission on the amount of sales she makes. If she makes $x in sales, she earns 10 % of x.
0.10x
Combining these two pieces of information, we can write the equation for f(x).
f(x)=0.10x+450
Because x represents the sales Lindsay makes a week, it cannot be a negative number and the domain of the function is the set of all real numbers greater than 0. Since she earns at least $450 a week, the range is the set of all real values of f(x) greater than 450.
Domain:& x≥ 0
Range:& f(x) ≥ 450
Rule for g(x)
Let's determine the rule of g(x) by looking at the graph! We can see that the y-intercept is at y=300. So far we have the following equation.
g(x)=mx+300
To find the slope, let's identify another point on the line!
Now let's substitute (0,300) and (4000,900) in the Slope Formula!
m = y_2 - y_1/x_2 - x_1
m = 900 - 300/4000 - 0
m = 600/4000
m = 0.15
The slope is 0.15, so we can write the final rule for g(x).
g(x)=0.15x+300
From the graph, we can see that the domain is the set of all real numbers greater than 0. The range is the corresponding output values, which is all real numbers of g(x) greater than or equal to 300.
Domain:& x≥ 0
Range:& g(x) ≥ 300
Comparison of Domains, Ranges, Slopes, and y-intercepts
Let's compare the slopes and y-intercepts of the functions visually by using a table.
| Feature
|
f(x)
|
g(x)
|
| Slope
|
0.10
|
0.15
|
| y-intercept
|
450
|
300
|
Now, we can think about each feature one at a time.
- The slope of g(x) is greater than the slope of f(x).
- The y-intercept of f(x) is greater than the y-intercept of g(x).
- The functions have the same domain: the set of all real numbers greater than 0.
- The ranges differ but are both unbounded by a maximum value.
- The initial value of f(x) is 450 and the initial value of g(x) is 300.
Graphing the Functions
Let's graph the functions on one coordinate plane!
We can tell that the graphs intersect at x=3000.
Comparison of the Benefits and Drawbacks
Using Plan F, Lindsay would have a higher base salary each week. This means that, even if she has a hard week where she made fewer sales, she would still earn at least $450. Having a hard week in sales while using Plan G would give her less money. On the other hand, using Plan G and making a lot of sales, let's say, $2000 a week, would give her additional
0.15 * 2000= $300
in commission. However, using Plan F and making sales of $2000 dollars in a week would give her only
0.10 * 2000= $200
extra in commission, which is quite a significant difference.
Lindsay's Decision
To make the right decision, Lindsay should ask her employer what is the average amount they sell each week. The lines on the graph intersect at x=3000. This tells us that, if Lindsay makes less than $3000 in sales each week, then it is more beneficial for her to choose Plan F. However, if the employer told her that the average sales are more than $3000 a week, then she should definitely choose Plan G.
