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If f(x) = cosx  g(x) = x^2  find a if f(g(a)) = 1

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ryanomar | Student, Undergraduate | eNoter

Posted December 4, 2010 at 1:11 PM via web

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If f(x) = cosx  g(x) = x^2  find a if f(g(a)) = 1

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted December 4, 2010 at 1:13 PM (Answer #1)

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Now are given that f(x) = cos x  and g(x) = x^2.

If f(g(a)) = 1

=> f( a^2) = 1

=> cos a^2 = 1

=> a^2 = arc cos 1

=> a^2 = 0

=> a = 0

Therefore a is equal to zero.

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hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted December 4, 2010 at 1:12 PM (Answer #2)

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Given the functions:

f(x) = cosx.

g(x) = x^2.

We need to determine the value of (a) such that f(g(a)) = 1

First, let us determine the function f(g(x)).

f(g(x)) = f( x^2)

          = cos(x^2).

==> f(g(x)) = cos(x^2).

Now we will substitute with  x = a.

==> f(g(a)) = cos(a^2)

But, given that f(g(a)) = 1

==> f(g(a)) = cos(a^2) = 1

Then a^2 = 0

==> a= 0.

Then the value of a is 0.

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giorgiana1976 | College Teacher | Valedictorian

Posted December 4, 2010 at 4:32 PM (Answer #3)

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To determine the value of a, we'll have to determine first the composition of the given functions f and g.

f(g(x)) is the result of composing f and g:

(fog)(x) = f(g(x))

To determine the expression of the composed function, we'll substitute x by g(x) and we'll get:

f(g(x)) = cos g(x)

Now, we'll substitute g(x) by it's expression:

f(g(x)) = cos x^2

Since we know the expression of f(g(x)), we can determine f(g(a)):

f(g(a)) = cos a^2

But, from enunciation, f(g(a)) = 1, so:

cos a^2 = 1

a^2 = +/-arccos 1 + 2*k*pi

a^2 = 0 + 2*k*pi

a = +/-sqrt 2kpi

or

a = 0

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