Question

Let f(x) = x^2and g(x) = x – 6, find:a. (fog)(x) =b. (gof)(x) =c. (fog)( - 1) = d. (gof)( - 1) =

Answer 1

Answer:

Given that,

$\begin{gathered} f(x)=x^2 \\ g(x)=x-6 \end{gathered}$

To find,a. (fog)(x)

b. (gof)(x)

c. (fog)( - 1)

d. (gof)( - 1)

we know that,

The composition of a function is an operation where two functions say f and g generate a new function say h in such a way that h (x) = g (f (x)). It means here function g is applied to the function of x that is f(x).

It is represented as (gof)(x), i.e)

$(g\circ f)(x)$

a)(fog)(x)

$(f\circ g)(x)=f(g(x))$$=f(x-6)$

Put x=x-6 in the function f(x),

$=(x-6)^2$

we get,

$(f\circ g)(x)=(x-6)^2-----(1)$

b)(gof)(x)

$(g\circ f)(x)=g(f(x))$$=g(x^2)$

Put x=x^2 in the function g(x),

$=x^2-6$

we get,

$(g\circ f)(x)=x^2-6----(2)$

c) (fog)( - 1)

using equation (1), Put x=-1, we get

$(f\circ g)(-1)=(-1-6)^2$$=(-7)^2=49$$(f\circ g)(-1)=49$

d) (gof)( - 1)

using the equation (2), Put x=-1, we get

$(g\circ f)(-1)=(-1)^2-6$$=1-6=-5$

$(g\circ f)(-1)=-5$