Literally complex functions

Algebra Level 3

Four functions are defined as follows:

$f(x) = x(x+1)$

$g(x) = x^2+3$

$h(x) = g(x) \cdot g^2(x)$

$h'(x) = g(x) + g^2(x)$

Find the value of $k$ given that $kf^2(2) = h(i) \cdot h'(i)$ where $i$ denotes the imaginary unit.

Note: $f^n(x) = f^{n-1} f(x)$

$f^n(x)$ is NOT $(f(x))^n$

The answer is 3.

1 solution

Noel Lo
Apr 26, 2015

$g(i) = i^2+3 =-1+3 = 2$

$g^2(i) = g(g(i)) = g(2) = 2^2+3 = 4+3 = 7$

$kf^2(2) = h(i) \cdot h'(i)$

$kf(f(2)) = (g(i) \cdot g^2(i)) \cdot (g(i) + g^2(i))$

$kf(2 \cdot 3) = (2 \cdot 7) \cdot (2+7)$

$kf(6) =14 \cdot 9$

$k(6 \cdot 7) = 126$

$42k =126$

$k = \frac{126}{42} = \boxed{3}$

This question is practically the same one as your previous question. However this time you had that note. Thanks for that one.

Utkarsh Singh - 6 years, 1 month ago

Yes I know Utkarsh Singh! I updated it because it was presented in a rather confusing way. I figured that it was confusing as no one could answer the question earlier, yeah. Just to clarify.

Noel Lo - 6 years, 1 month ago

Literally complex functions

The answer is 3.

1 solution

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