Question

If a(x) = 3x + 1 and b(x)= √x-4 , what is the domain of (b*a)(x)

Answer 1

a(x) = 3x + 1
b(x) = √(x-4)

Therefore
$(b \circ a) (x) = \sqrt{3x+1-4} =\sqrt{3x-3}$
This composition of functions is only defined as a real number when x≥0.
Therefore its domain is x = [1,∞]

Answer: [1,∞]

Answer 2

Answer: The correct option is C. The domain of  $(b\circ a)(x)$  is  $[0,\infty)$.

Explanation:

It is given that,

$a(x)=3x+1$

$b(x)=\sqrt{x-4}$

First we have to find the composite function $(b\circ a)(x)$.

$(b\circ a)(x)=b(a(x))$

$(b\circ a)(x)=b(3x+1)$

$(b\circ a)(x)=\sqrt{3x+1-4}$

$(b\circ a)(x)=\sqrt{3x-3}$

We know that a root function is defined for only positive values therefore the composite function is defined if,

$3x-3\geq 0$

$3x\geq 3$

$x\geq 1$

The function is defined for all the values of x which are greater than or equal to 1. Therefore the domain of  $(b\circ a)(x)$  is  $[0,\infty)$ and option C is correct.