Functions and Algebra question

Question

Given that $f(t) = 2t^2 + 4$, $f(4x + 2) = 2(16x^2 + 4) = 32x^2 + 12$.

This is incorrect and I have no idea why, the correct answer is

$2(16x^2 + 16x + 4) = 32x^2 + 32x + 12$. I can't work out where $16x$ came from.

Answer 1

Guide:

Evaluate $$2(4x+2)^2+4$$ again.

Note that $(A+B)^2 = A^2+2AB+B^2$.

Answer 2

You have to square the whole thing! What is $(a+b)^2$?

Here is how to deduce the correct answer, see if you can justify each step:

$(4x+2)= t \implies f(t)=2t^2+4=2(4x+2)^2+4=2(16x^2+16x+4)+4$.

Answer 3

\begin{eqnarray*} f(t)&=&2t^2+4 \\ f( \color{blue}{4x+2})&=& 2(\color{blue}{4x+2})^2+4. \end{eqnarray*}

Answer 4

You’ve made the mistake of thinking that $(4x+2)^2$ is $16x^2 +4$.

That would work if the “$+$” were “$\times$”, but exponents don’t distribute over terms in a sum.

Instead, remember that $$(4x+2)^2=(4x+2)(4x+2)$$ $$=4x(4x+2) +2(4x+2)$$ $$=(16x^2+8x) +(8x+4)$$ $$=16x^2 + 16x +4$$ and proceed from there.