the monthly base salary of a shoes sales person is birr 900. she has a commission of 2% on all sales oer birr 10,000 during the the month. if the monthly sales are birr 15,000 or more, she receives birr 500 bonus. if x represents the monthly sales in birr and f(x) represents income in birr, express f(x) in terms of x and discuss the continuity of f on [0, 25000]!
for x in [0,10000), f(x) = 900
for x in [10000,15000), f(x) = 900+.02(x-10000)
for x in [15000,25000] f(x) = 900+.02(x-10000) + 500
I think you can discuss the continuity. If not, draw the graph and it will become clear.
-6000
-60000
5500
Oh, commission and bonus, huh? This shoe salesperson is getting quite the deal! Now, let's break down the function f(x) to find the income in terms of x.
For monthly sales x < 10,000, there is no commission. So, f(x) = 900.
For monthly sales x ≥ 10,000, the commission is 2%. Hence, the commission is 0.02(x - 10,000). However, we also have the bonus of 500 if the sales are 15,000 or more. So, f(x) = 900 + 0.02(x - 10,000) + 500.
Now let's discuss the continuity of f on [0, 25000].
From 0 to 10,000, f(x) is equal to 900, which is a constant. So, f is continuous on this interval.
From 10,000 onwards, we have f(x) = 900 + 0.02(x - 10,000) + 500. This is a linear equation, with a slope of 0.02. Therefore, f(x) is still continuous on this interval.
So, in conclusion, the function f(x) = 900 for x < 10,000, and f(x) = 900 + 0.02(x - 10,000) + 500 for x ≥ 10,000. And f remains continuous on the interval [0, 25000]. Hope that clarifies things!
To express f(x) in terms of x, we need to consider the different scenarios based on the monthly sales amount.
1. If x ≤ 10,000:
The salesperson receives a commission of 2% on all sales. So f(x) = 900 + 0.02x.
2. If x > 10,000 and x < 15,000:
The salesperson receives a bonus of 500 birr, in addition to the base salary and commission. So f(x) = 900 + 0.02x + 500 = 1400 + 0.02x.
3. If x ≥ 15,000:
The salesperson receives both the bonus of 500 birr and the commission on sales over 10,000 birr. So f(x) = 900 + 0.02(10,000) + 0.02(x - 10,000) + 500 = 1100 + 0.02x.
Now, let's discuss the continuity of f(x) on the interval [0, 25,000].
For continuity, there are three conditions that need to be met:
1. The function must be defined for all values in the interval [0, 25,000].
2. The function must be finite for all values in the interval [0, 25,000].
3. The function must have a limit as x approaches any value in the interval [0, 25,000].
In this case, f(x) is defined for all values of x between 0 and 25,000 as we have expressed it for different scenarios. Additionally, the function is finite for all values of x in this interval.
To check the limit as x approaches any value in the interval, we need to analyze the three scenarios separately.
1. If x ≤ 10,000:
As x approaches any value in [0, 10,000], f(x) = 900 + 0.02x and since this is a linear function, it is continuous.
2. If x > 10,000 and x < 15,000:
As x approaches any value in the interval (10,000, 15,000), f(x) = 1400 + 0.02x, which is also a linear function. Hence, it is continuous in this interval.
3. If x ≥ 15,000:
As x approaches any value in the interval [15,000, 25,000], f(x) = 1100 + 0.02x, which is again a linear function. Therefore, it is continuous in this interval.
Considering all the scenarios, f(x) is continuous on the interval [0, 25,000].