Question

Math equation help please. 20 points!

Answer 1

Answer:

(a) we are given two functions f(x) and g(x) as:

$f(x)=\dfrac{1}{x-4}$ and $g(x)=\dfrac{4x+1}{x}$

Now we find out the composition maps fog and gof as:

$fog=f(g(x))\\\\=f(\dfrac{4x+1}{x})\\\\=\dfrac{1}{\dfrac{4x+1}{x}-4 }\\ \\=\dfrac{x}{4x+1-4x}\\ \\=x$

Hence, $fog=x$

similarly we find gof.

$gof(x)=g(f(x))\\\\=g(\dfrac{1}{x-4})\\\\=\dfrac{4(\dfrac{1}{x-4})+1 }{\dfrac{1}{x-4}}\\ \\=\dfrac{\dfrac{4+x-4}{x-4} }{\dfrac{1}{x-4}}\\ \\=x$

hence,  $gof=x$

as fog=gof=x this means that f and g are inverse of each other.

(b) the domain of f is given by: (-∞,4)∪(4,∞)

since, f is defined all over the real line except at 4 ; at 4 the function is not defined as the denominator is zero.

also g is defined everywhere except at 0; since at 0 denominator is 0.

hence, domain of g(x) is (-∞,0)∪(0,∞).

domain of the composition fog is the domain of the function g(x) ( as for defining fog we need to evaluate function g(x) first and then f(g(x)) ).

and domain of function gof is equal to domain of function f(x) ( as for defining gof we need to evaluate function f(x) first and then g(f(x)) ).

hence domain of fog=(-∞,0)∪(0,∞).

and domain of gof=(-∞,4)∪(4,∞)