Answer:
<u>Question 1</u>
The function 
is one-to-one, so it does have an inverse.
The inverse of +6 is -6, so 
Therefore, g(x) is the inverse of f(x).
<u>Question 2</u>
The function 
is one-to-one, so it does have an inverse.
To find the inverse, replace f(x) with y:
Rearrange the equation to make x the subject:
Replace x with 
and y with x:
Therefore, g(x) is the inverse of f(x).
Answer:
Fast ball challenge
Step-by-step explanation:
Given
Slow Ball Challenge
Fast Ball Challenge
Required
Which should he choose?
To do this, we simply calculate the expected earnings of both.
Considering the slow ball challenge
First, we calculate the binomial probability that he hits all 7 pitches
Where
--- pitches
--- all hits
--- probability of hit
So, we have:
Using a calculator:
--- This is the probability that he wins
i.e.
The probability that he lose is:
---- Complement rule
The expected value is then calculated as:
Using a calculator, we have:
Considering the fast ball challenge
First, we calculate the binomial probability that he hits all 3 pitches
Where
--- pitches
--- all hits
--- probability of hit
So, we have:
Using a calculator:
--- This is the probability that he wins
i.e.
The probability that he lose is:
---- Complement rule
The expected value is then calculated as:
Using a calculator, we have:
So, we have:
-- Slow ball
--- Fast ball
<em>The expected earnings of the fast ball challenge is greater than that of the slow ball. Hence, he should choose the fast ball challenge.</em>
Given plane Π : f(x,y,z) = 4x+3y-z = -1
Need to find point P on Π that is closest to the origin O=(0,0,0).
Solution:
First step: check if O is on the plane Π : f(0,0,0)=0 ≠ -1 => O is not on Π
Next:
We know that the required point must lie on the normal vector <4,3,-1> passing through the origin, i.e.
P=(0,0,0)+k<4,3,-1> = (4k,3k,-k)
For P to lie on plane Π , it must satisfy
4(4k)+3(3k)-(-k)=-1
Solving for k
k=-1/26
=>
Point P is (4k,3k,-k) = (-4/26, -3/26, 1/26) = (-2/13, -3/26, 1/26)
because P is on the normal vector originating from the origin, and it satisfies the equation of plane Π
Answer: P(-2/13, -3/26, 1/26) is the point on Π closest to the origin.
Proving a relation for all natural numbers involves proving it for n = 1 and showing that it holds for n + 1 if it is assumed that it is true for any n.
The relation 2+4+6+...+2n = n^2+n has to be proved.
If n = 1, the right hand side is equal to 2*1 = 2 and the left hand side is equal to 1^1 + 1 = 1 + 1 = 2
Assume that the relation holds for any value of n.
2 + 4 + 6 + ... + 2n + 2(n+1) = n^2 + n + 2(n + 1)
= n^2 + n + 2n + 2
= n^2 + 2n + 1 + n + 1
= (n + 1)^2 + (n + 1)
This shows that the given relation is true for n = 1 and if it is assumed to be true for n it is also true for n + 1.
<span>By mathematical induction the relation is true for any value of n.</span>
45 minutes equal to 2700 seconds.
