Question

Given that f(x) = 6x - 5 g(x) = 3x + 4 and h(x) = 4x – 6

Answer 1

Answer:

(i) $g(-2)=-2$

(ii) $g[h(x)]=12x-14$

(iii) $f[g(2)]=55$

(iv) $(g\circ h)(2)=10$

(v) $h^{-1}(11)=\frac{17}{4}$

Step-by-step explanation:

The given functions are

$f(x)=6x-5$

$g(x)=3x+4$

$h(x)=4x-6$

(i) Find g(-2).

Substitute x=-2 in g(x).

$g(-2)=3(-2)+4\Rightarrow -6+4=-2$

(ii) Find g[h(x)]

$g[h(x)]=g(4x-6)$                 $(h(x)=4x-6)$

$g[h(x)]=3(4x-6)+4$           $(g(x)=3x+4)$

$g[h(x)]=12x-18+4$

$g[h(x)]=12x-14$

(iii) Find f[g(2)].

$f[g(2)]=f[3(2)+4]$           $(g(x)=3x+4)$

$f[g(2)]=f(6+4)$

$f[g(2)]=f(10)$

$f[g(2)]=6(10)-5$           $f(x)=6x-5$

$f[g(2)]=55$

iv) Find gᴏh(2).

$(g\circ h)(2)=g[h(2)]$

$(g\circ h)(2)=12(2)-14$           (From part (ii) we get $g[h(x)]=12x-14)$

$(g\circ h)(2)=10$

(v) Find $h^{-1}(11)$

First find $h^{-1}(x)$.

$h(x)=4x-6$

$y=4x-6$

$x=4y-6$

$x+6=4y$

Divide both sides by 4.

$\frac{x+6}{4}=y$

$h^{-1}(x)=\frac{x+6}{4}$

Substitute x=11 in the inverse function.

$h^{-1}(11)=\frac{11+6}{4}$

$h^{-1}(11)=\frac{17}{4}$