Question 1 :
Let f = {(-1, 3), (0, -1), (2, -9)} be a linear function from Z into Z . Find f(x).
Answer :
Let the linear function be f(x) = ax + b.
From the ordered pair (-1, 3), x = -1, f(x) = 3.
3 = a(-1) + b
3 = -a + b ----(1)
From the ordered pair (0, -1), x = 0, f(x) = -1.
-1 = a(0) + b
-1 = b
Substitute b = -1 in (1).
3 = -a + (-1)
3 = -a - 1
Add a to each side.
3 + a = -1
Subtract 3 from each side.
a = -4
So, the linear function is f(x) = -4x - 1.
Question 2 :
In electrical circuit theory, a circuit C(t) is called a linear circuit if it satisfies the superposition principle given by C(at_{1} + bt_{2}) = aC(t_{1}) + bC(t_{2}), where a, b are constants. Show that the circuit C(t) = 3t is linear.
Answer :
Take two points t_{1} and t_{2} from domain of C(t).
C(at_{1}) = aC(t_{1})
C(at_{2}) = aC(t_{2})
It is given that C(t) = 3t.
C(t) = 3t
C(at_{1 }+ bt_{2}) = 3(at_{1 }+ at_{2})
C(at_{1 }+ bt_{2}) = 3at_{1 }+ 3at_{2}
C(at_{1 }+ bt_{2}) = a(3t_{1}) + a(3t_{2})
C(at_{1 }+ bt_{2}) = aC(t_{1}) + aC(t_{2})
Superposition principle is satisfied.
Hence c(t) = 3t is linear.
Question 3 :
The cost of a school banquet is $95 plus $15 for each person attending. Write a linear function that gives total cost of the number of people attending. What is the cost for 77 people?
Solution :
Let f(x) be the total cost and x be the number of persons attending the banquet.
Linear function that gives total cost :
f(x) = 95 + 15x
To find the cost for 77 people, substitute x = 77.
f(77) = 95 + 15(77)
= 95 + 1155
= 1250
So, the total cost of attending 77 people is $1250.
Question 4 :
A manufacturer produces 80 units of a particular product at a cost of $ 220000 and 125 units at a cost of $ 287500. Assuming the cost curve to be linear, find the cost of 95 units.
Solution :
Because the cost curve is linear, the function which best fits the given information will be a linear-cost function.
y = Ax + B
y ----> Total cost
x ----> Number of units
Target : To find the value of y when x = 95.
From the question, we have
x = 80 and y = 220000
x = 75 and y = 287500
Substitute the above values for x and y in 'y = Ax + B'.
220000 = 80A + B
287500 = 75A + B
Solve for x and y.
A = 1500 and B = 100000
The linear cost function :
y = 1500x + 100000
Substitute x = 95.
y = 1500x + 100000
y = 1500x95 + 100000
y = 142500 + 100000
y = 242500
So, the cost of 95 units is $242500.
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