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Quantitative Ability and Data Interpretation Question on Logarithms

Let f(x) = 4(x2+x), g(x)= √x+1, h(x) = g(f(x)). Find the domain and range of h(x).

A

(-∞, ∞) and (-∞, ∞)

B

(0, ∞) and (-∞, ∞)

C

(-∞, ∞) and (0, ∞)

D

(0, ∞) and (0, ∞)

E

None of these

Answer

(-∞, ∞) and (0, ∞)

Explanation

Solution

f(x)=4(x2+x)f(x) = 4(x^2+x)

h(x)=g(f(x))=g(4(x2+x)+1=4x2+4x+1h(x) = g(f(x)) = g(\sqrt{4(x^2+x)+1} = \sqrt{4x^2+4x+1}

= (2x+1)2=∣2x+1∣\sqrt{(2x+1)^2} = |2x+1|

So, the domain of h(x)h(x) is (−∞,∞)(-∞, ∞).

The range of h(x)=(0,∞)h(x) = (0, ∞) as the output will not contain any negative values.

Hence, option C is the correct answer.