domain of f(x)=((3x+2))/(sqrt(x^2-7x)) — step-by-step solution | Explain MAths

domain of f(x)=((3x+2))/(sqrt(x^2-7x))

We save the definition:

f (x) = \frac{3 x + 2}{\sqrt{x^{2} - 7 x}}

1) Let's find the domain of

\frac{3 x + 2}{\sqrt{x^{2} - 7 x}}

For the expression

\frac{3 x + 2}{\sqrt{x^{2} - 7 x}}

to be defined, we must have:

x^{2} - 7 x > 0

You can factorize the expression:

x^{2} - 7 x

into:

x \cdot (x - 7)

We study the sign of each factor of the product

⇥The expression

x

is zero at

x = 0

, it is negative before and positive after

⇥The expression

x - 7

is zero at

x = 7

, it is negative before and positive after

2) Let's draw up the sign table:

x	$-∞$		$0$		$7$		$+∞$
$x$		-	0	+		+
$x - 7$		-		-	0	+
$x^{2} - 7 x$		+	0	-	0	+

▶Domain of definition:

x \in (-∞, 0) \cup (7, +∞)