# How do you find (f*g)(-1) given f(x)=x^2-1 and g(x)=2x-3 and h(x)=1-4x?

Sep 12, 2017

$\left(f \cdot g\right) \left(- 1\right) = 24$

#### Explanation:

I personally find it easier to think of
$\textcolor{w h i t e}{\text{XXX}} \left(f \cdot g\right) \left(x\right)$ and $\left(f \cdot g\right) \left(- 1\right)$
as $f \left(g \left(x\right)\right)$ and f(g(-1) respectively

Given
$\textcolor{w h i t e}{\text{XXX}} f \left(\textcolor{b l u e}{x}\right) = {\textcolor{b l u e}{x}}^{2} - 1$
then
after simply substituting $\textcolor{b l u e}{g \left(x\right)}$ for $x$
$\textcolor{w h i t e}{\text{XXX}} f \left(\textcolor{b l u e}{g \left(x\right)}\right) = {\left(\textcolor{b l u e}{g \left(x\right)}\right)}^{2} - 1$

Now since we are also given that
color(white)("XXX")color(blue)(g(color(magenta)x)=2color(magenta)x-3
after substituting $\textcolor{b l u e}{2 \textcolor{m a \ge n t a}{x} - 3}$ for $\textcolor{b l u e}{g \left(x\right)}$ above
we get
$\textcolor{w h i t e}{\text{XXX}} f \left(\textcolor{b l u e}{g \left(\textcolor{m a \ge n t a}{x}\right)}\right) = {\left(\textcolor{b l u e}{2 \textcolor{m a \ge n t a}{x} - 3}\right)}^{2} - 1$

Expanding (and dropping the color coding for a bit)
$\textcolor{w h i t e}{\text{XXX}} f \left(g \left(x\right)\right) = 4 {x}^{2} - 12 x + 8$

...or we could re-color code the variable $\textcolor{red}{x}$ to make this appear as
$\textcolor{w h i t e}{\text{XXX}} f \left(g \left(\textcolor{red}{x}\right)\right) = 4 {\textcolor{red}{x}}^{2} - 12 \textcolor{red}{x} + 8$

Since we are asked for $\left(f \cdot g\right) \left(\textcolor{red}{- 1}\right)$
which can be written as $f \left(g \left(\textcolor{red}{- 1}\right)\right)$

we can substitute $\left(\textcolor{red}{- 1}\right)$ in place of $\textcolor{red}{x}$ in our definition of $f \left(g \left(\textcolor{red}{x}\right)\right)$
$\textcolor{w h i t e}{\text{XXX}} f \left(g \left(\textcolor{red}{- 1}\right)\right) = 4 \cdot {\left(\textcolor{red}{- 1}\right)}^{2} - 12 \cdot \left(\textcolor{red}{- 1}\right) + 8$

$\textcolor{w h i t e}{\text{XXXXXXXX}} = 4 + 12 + 8$

$\textcolor{w h i t e}{\text{XXXXXXXX}} = 24$