Question

Determine the domain and range of (g ○ f)(x) if f of x is equal to 4 over the quantity x squared minus 4 end quantity and g(x) = x + 2.

Answer 1

For the function (gof)(x) the Domain is {x∈ R| ,x ≠ –2, 2} , the Range is {(–∞, 1)∪(2, ∞)} .

In the question ,

it is given that

the function f(x) is given as "4 over the quantity x squared minus 4" ,

which means

f(x)=4/(x²-4)

and g(x)=x+2

we know ,

(gof)(x) = g(f(x))= f(x) + 2

= 4/(x²-4) + 2

taking LCM as (x²-4) and solving further ,

we get

= $\frac{4+2x^{2} -8}{x^{2} -4}$

= (2x²-4)/(x²-4)

the fraction is defined only when the denominator is non zero,

So for Domain

x²-4 ≠ 0

x² ≠ 4

x ≠ +2,-2

So, the domain is all real numbers except the numbers 2,-2 .

For Range , the graph of the function (gof)(x) is plotted below ,

From the graph we can see that the function does not have any value from y=1 to 2 .

So, the Range will be R={(∞, 1)∪(2, ∞)}.

Therefore , for the function (gof)(x) ,the Domain is {x∈ R| ,x ≠ –2, 2} , the Range is R:{(–∞, 1)∪(2, ∞)} .

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