The domain of [tex]b\,\circ \,a\,(x)[/tex] is [tex][1, +\infty)[/tex].
How to apply composition between functions
The composition of two functions consists in replacing the variable of the former function ([tex]b(x)[/tex]) by the latter function ([tex]a(x)[/tex]). If we know that [tex]b(x) = \sqrt{x-4}[/tex] and [tex]a(x) = 3\cdot x +1[/tex], then the composite function:
[tex]b\,\circ\,a \,(x) = \sqrt{(3\cdot x+1)-4}[/tex]
[tex]b\,\circ\,a\,(x) = \sqrt{3\cdot x-3}[/tex] (1)
Since the composite function is a square root function, then the domain fulfill the following inequation:
[tex]3\cdot x -3 \ge 0[/tex]
[tex]x \ge 1[/tex]
Hence, the domain of [tex]b\,\circ \,a\,(x)[/tex] is [tex][1, +\infty)[/tex]. [tex]\blacksquare[/tex]
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