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Given that \(a > 0\), if \(f(g(a)) = 8\), where \(f(x) = x^2 + 8\) and \(g(x) = x^2 - 4\), what is the value of \(a\)?

 Apr 6, 2021
 #1
avatar+56 
+2

Given that a > 0, if f(g(a)) = 8, where f(x) = x^2 + 8 and g(x) = x^2 - 4, what is the value of a?

 

Substitute in g(x)

f(g(a)) = f(a^2 - 4)

f(a^2 - 4) = (a^2 - 4)^2 + 8

 

We bring that back into the original equation

(a^2 - 4)^2 + 8 = 8

(a^2 - 4)^2 = 0

a^2 - 4 = 0

a^2 = 4, a = 2

 

Answer: a = 2

 

Check:

f(0) = 0^2 + 8 = 8

 

:)

 Apr 6, 2021
 #2
avatar+33757 
+1

Given that    f( g(a)  ) = 8    then the thing in the parentheses must be  0

                               because  g(a)^2  + 8 = 8      then g(a)^2 = 0         g(a) = 0

 

so   g(a) must = 0

           0 =a^2-4

             a^2 = 4     a = 2   (since we only want positives)

 Apr 6, 2021

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