Question

comparing graphs of radical functions
which of the following graphs have the same domain? choose three correct answers.
$y = \sqrt{x + 4}$
$y = \sqrt{x}$
$y = 4\sqrt{x}$
$y = \sqrt{x} + 4$

Explanation:

Step1: Find domain of \( y = \sqrt{x + 4} \)

For a square root function \( \sqrt{f(x)} \), the domain requires \( f(x) \geq 0 \). So \( x + 4 \geq 0 \implies x \geq - 4 \).

Step2: Find domain of \( y = \sqrt{x} \)

For \( \sqrt{x} \), \( x \geq 0 \) (since the expression under the square root must be non - negative).

Step3: Find domain of \( y = 4\sqrt{x} \)

For \( 4\sqrt{x} \), the expression under the square root is \( x \), so \( x \geq 0 \) (because the square root of a negative number is not a real number in the context of real - valued functions).

Step4: Find domain of \( y=\sqrt{x}+4 \)

For \( \sqrt{x}+4 \), the expression under the square root is \( x \), so we need \( x \geq 0 \) (the addition of 4 does not affect the domain of the square root part).

The functions \( y = \sqrt{x} \), \( y = 4\sqrt{x} \), and \( y=\sqrt{x}+4 \) all have the domain \( x\geq0 \), while \( y = \sqrt{x + 4} \) has a domain of \( x\geq - 4 \).

Answer:

The three graphs with the same domain are:

• \( y = \sqrt{x} \)
• \( y = 4\sqrt{x} \)
• \( y=\sqrt{x}+4 \)