QuestionQuestion 10 (1 point) Find $g \circ f$ and the domain of the composite function. $f(x)=x^{2}+4, g(x)=\sqrt{x}$ a $\sqrt{(x-4)^{4}}$ Domain of $g \circ f$ : all real numbers $x$ b $(x-4)^{4}$ Domain of $g \circ f$ : all real numbers $x$ c $\sqrt{x^{2}+4}$ Domain of $g \circ f$ : all real numbers $x$ d $\quad(x+4)^{4}$ Domain of $g \circ f$ : all real numbers $x$ e $\sqrt{(x+4)^{4}}$ Domain of $g \circ f$ : all real numbers $x$

Studdy Solution

STEP 1

1. The composite function $g \circ f$ is defined as $g(f(x))$ .
2. The domain of a composite function $g \circ f$ consists of all $x$ such that $f(x)$ is in the domain of $g$ .

STEP 2

1. Determine the expression for $g \circ f$ .
2. Determine the domain of $g \circ f$ .

STEP 3

First, find the expression for $g \circ f$ . This means substituting $f(x)$ into $g(x)$ :
Given: $f(x) = x^2 + 4$ $g(x) = \sqrt{x}$
The composite function $g \circ f$ is: $g(f(x)) = g(x^2 + 4) = \sqrt{x^2 + 4}$

STEP 4

Determine the domain of $g \circ f$ .
The function $g(x) = \sqrt{x}$ is defined for $x \geq 0$ . Therefore, $f(x)$ must satisfy: $x^2 + 4 \geq 0$
Since $x^2 + 4$ is always positive for all real numbers $x$ (as $x^2 \geq 0$ and adding 4 makes it strictly positive), the domain of $g \circ f$ is all real numbers.
The correct answer is: c $\sqrt{x^{2}+4}$ with Domain of $g \circ f$ : all real numbers $x$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms