Question

1. The domain of the function h(x) = ln(x + 3) + x2 − 4 is

   (A) {x∈R| −3<x<−2or−2<x<0}
   (B) {x∈R| −3<x<−2or−2<x≤0}
   (C) {x∈R| −3≤x<−2or−2<x≤0}
   (D) {x∈R| −3<x<−2or0≤x<2}
   (E) {x∈R| −3<x<−2or(x≥0andx̸=2)} (F) None of (A) - (E)

Answer 1

Step 1)

we have to find the domain of the function h(x) = ln(x+3) + x² - 4

Let's assume that f(x) = ln(x+3) and g(x) = x² - 4

Hence we can write h(x) = f(x) + g(x)

Let's assume that domain of a function h(x) = D, domain of function f(x) = D₁ and domain of function g(x) = D₂ hence we can write,

Step 2)

we have f(x) = ln(x+3)

we know that domain of a function f(x) is set of input values for which f(x) is real and defined

we know that ln(x) is real and defined for every values of x such that x > 0

Hence we can write f(x) = ln(x+3) is real and defined for every values of x such that x + 3 > 0

we can write domain of f(x) is given by,

In interval notation we can write domain of f(x) is,

Step 3)

we have g(x) = x² - 4

we know that domain of a function g(x) is set of input values for which g(x) is real and defined

we can see that g(x) is a polynomial of degree 2 which is real and defined for all values of x hence we can say that domain of g(x) is given by,

Step 4)

we have,

Hence we can write,

we know that,

Hence,

Hence we can say that domain of h(x) = ln(x+3) + x² - 4 is given by,

we can also write,

Hence we can say that right choice is F) none of (A) - (E) as domain of h(x) is,