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3 years ago

Use the identity

Mathematics

1 answer:

dezoksy [38]3 years ago

5 0

The sum of the cubes of two numbers is 52

Explanation:

Let a and b be two numbers.

It is given that the sum of the two numbers is 4 and the product of the two numbers is 1

Thus, Rewriting it in the expression form, we have,

$a+b=4$

$ab=1$

To determine the sum of the cubes of two numbers, let us substitute these values in the identity $a^3+b^3=(a+b)^3-3ab(a+b)$ , we get,

$a^3+b^3=(4)^3-3(1)(4)$

$=64-12$

$=52$

Thus, the sum of the cubes of two numbers is 52

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Find an equation of the tangent line to y = 2 sin x at x = _/6

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4 years ago

Find the composite functions (f ∘ g) and (g ∘ f). what is the domain of each composite function? \[ \begin{array}{rcl} f(x) &amp

Burka [1]

Always remember two steps while finding the domain of composite function.

1) <em>First find the domain of inside/ input function( A common mistake is to skip this point).</em>

2) <em>Find the domain of new function after performing the composition.</em>

In our case, the given functions are

$f(x)=\frac{3}{x}$ , $g(x)= x^{2} -81$

Now,

1) (f ° g)(x)

= f[g(x)]

=f[x²-81]

$= \frac{3}{x^{2}-81}$ (this is (f ° g)(x) function)

Now, its domain

First find the domain of input function which is x²-81, its domain is the set of all real numbers.

domain of new function after performing composition which is $\frac{3}{x^{2}-81}$ .

x²-81=0 ⇒ (x-9)(x+9)=0

(x-9)=0 , (x+9)=0

x=9 , x=-9 (exclude these points from the domain because anything/0 does not exist in math)

So, the Domain of Composite function (f °g)(x) is the set of all real numbers except x=9, x=-9.

Domain= (-infinity, -9)U(-9, 9)U(9, infinity)

2) (g °f)(x)

= g[f(x)]

= g[3/x]

= (3/x)² -81

= 9/x² -81

$=\frac{9-81x^{2}}{x^{2}}$

First find the domain of input function 3/x which is set of all real numbers except x=0

$\frac{9-81x^{2}}{x^{2}}$

Domain of above function after performing composition is set of all real numbers except x=0

So, the domain of (g° f)(x) is the set of all real numbers except x=0.

Domain= (-infinity, 0)U(0, infinity)

6 0

4 years ago