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Note that f(x) as given is <em>not</em> invertible. By definition of inverse function,
which is a cubic polynomial in with three distinct roots, so we could have three possible inverses, each valid over a subset of the domain of f(x).
Choose one of these inverses by restricting the domain of f(x) accordingly. Since a polynomial is monotonic between its extrema, we can determine where f(x) has its critical/turning points, then split the real line at these points.
f'(x) = 3x² - 1 = 0 ⇒ x = ±1/√3
So, we have three subsets over which f(x) can be considered invertible.
• (-∞, -1/√3)
• (-1/√3, 1/√3)
• (1/√3, ∞)
By the inverse function theorem,
where f(a) = b.
Solve f(x) = 2 for x :
x³ - x + 2 = 2
x³ - x = 0
x (x² - 1) = 0
x (x - 1) (x + 1) = 0
x = 0 or x = 1 or x = -1
Then can be one of
• 1/f'(-1) = 1/2, if we restrict to (-∞, -1/√3);
• 1/f'(0) = -1, if we restrict to (-1/√3, 1/√3); or
• 1/f'(1) = 1/2, if we restrict to (1/√3, ∞)
I'll explain it simply for you
1st question
Of course you know phythagoras theorm
You even wrote it up there
It states that the sum of the square of the two sides of an equilateral triangle is equal to the square of the hypotenuse
Where C is the hypotenuse
*NOTE* :
HYPOTENUSE is the greatest side in a triangle!!
And that's where your mistake is!
So you should take the greatest side as C
So in Q3. 7, 24 and 26 are the given numbers
You'll make the smaller two numbers a and b and the greatest number C
Using the Formula you'll solve the left side first which is
Then the right side which is
And if both are equal then it is a right triangle otherwise it isn't!
Let
a=7
b=24
c=26
a^2 + b^2
7^2 + 24^2
49 + 576 = 625
GREAT, Now the right side
26^2 = 676
Since they aren't equal it isn't a right angled triangle...
Then let
a=7.5
b=10
c=12.5
7.5^2 + 10^2
= 56.25 + 100
= 156.25
12.5^2 = 156.25
They are EQUAL
Therefore it is a right triangle too
Hopefully I helped
1st question
Of course you know phythagoras theorm
You even wrote it up there
It states that the sum of the square of the two sides of an equilateral triangle is equal to the square of the hypotenuse
Where C is the hypotenuse
*NOTE* :
HYPOTENUSE is the greatest side in a triangle!!
And that's where your mistake is!
So you should take the greatest side as C
So in Q3. 7, 24 and 26 are the given numbers
You'll make the smaller two numbers a and b and the greatest number C
Using the Formula you'll solve the left side first which is
Then the right side which is
And if both are equal then it is a right triangle otherwise it isn't!
Let
a=7
b=24
c=26
a^2 + b^2
7^2 + 24^2
49 + 576 = 625
GREAT, Now the right side
26^2 = 676
Since they aren't equal it isn't a right angled triangle...
Then let
a=7.5
b=10
c=12.5
7.5^2 + 10^2
= 56.25 + 100
= 156.25
12.5^2 = 156.25
They are EQUAL
Therefore it is a right triangle too
Hopefully I helped
Answer:
Step-by-step explanation:
Given
See attachment
Required
Determine the measure of
and
are on a straight line.
So:
--- angle on a straight line
Substitute known values
Collect like terms
