Question

Let $f$ and $g$ be functions from $R$ to $R$ defined as
$f(x) = \begin{cases} 7x^2+x-8, x \le 1 & \quad \text{} \\ 4x + 5, 1< x \le 7 & \quad \text{} \\ 8x+3, x > 7 & \quad \text{ } \end{cases} $ and $f(x) = \begin{cases} |x|, x <-3 & \text{} \\ 0, -3 \le x <2& \text{ } \\ x^2+4, x \ge 2& \text{ } \end{cases} $
Then,

• A. $(fog) (-3) = 8$
• B. $(fog) (9) = 683$
• C. $(gof) (0) = - 8$
• D. $(gof) (6) = 427$

Answer 1

Answer: (B)

We have $g(-3) = 0$
$\Rightarrow f \left(g\left(-3\right)\right)=f \left(0\right)=7\left(0\right)^{2}+0-8$
$=-8$
Now, $g\left(9\right)=9^{2}+4$
$=85$
$\Rightarrow f\left(g\left(9\right)\right)=f \left(85\right)=8\times85+3$
$=683$
Also, $f \left(0\right)=7\left(0\right)^{2}+0-8$
$=-8$
$\Rightarrow g\left(f \left(0\right)\right)=g\left(-8\right)=\left|-8\right|=8$
Also, $f \left(6\right)=4\times6+5$
$=29$
$\Rightarrow g\left(f \left(6\right)\right)=g\left(29\right)=\left(29\right)^{2}+4$
$=845$