For f(x)=3x+1 and g(x)=x^2-6, find (f o g)(x)
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Answer:
Step-by-step explanation:
We know that Megan can plant 1 garden bed in 8 hours. Let M represent Megan's rate. So:
We know that her mother can plant 2 garden beds in that time (8 hours). Let A represent the mother's rate. So:
We can reduce this to 1 flower bed every 4 hours.
We also know that Megan's father can plant 1 1/3 or 4/3 garden beds in 8 hours. Let D represent the father's rate. So:
We can reduce (4/3)/8 to 1/6.
We know that the three began working together. They worked together for 3 hours. So, after 3 hours, the amount of beds they planted all together is 3 hours times their respective rates. So, we can write the following expression:
We know that at this point, Megan's father left, leaving only Megan and her mother. We know that they worked together for another 30 minutes, or 1/2 of an hour. So, after this, they will have planted:
Garden beds.
Now, Megan's mother leaves, leaving only Megan. Let's let x represent the number of hours. So, we can write the last part of our expression:
We know that in the end, they planted 2 flower beds. So, our entire expression equals 2:
To find out how long it took Megan, we will solve for x.
Let's do each term individually:
First Term:
We have:
Make the fractions with common denominators. Our common denominator here is 24. So:
Add:
Multiply. So, our first term is:
Second Term:
We have:
Again, let's turn the fractions into fractions with common denominators so we can add them. The common denominator here is 8. So:
Add:
Multiply:
So, our equation is now:
Add on the left. Use the common denominator of 48. So:
Add:
Subtract 87/48 from both sides:
Let turn into a fraction with a denominator of 48. So:
Subtract:
Reduce the right using 3:
Multiply both sides by 8:
Reduce using 8. So, the time it will take Megan to finish planting the garden beds by herself is:
And we're done!