Suppose that the functions f and g are defined as follows. f(x)=(2)/(x)quad g(x)=x-3 Find (f)/(g). Then, give its domain using an interval or union of intervals Simplify your answers. ((f)/(g))(x)=◻ Domain of (f)/(g) :

Define functions: question_prompt: Find the function (f/g)(x) and determine its domain. Calculate (f/g)(x): Step 1: Define the functions f(x) and g(x). f(x) = 2/x, g(x) = x - 3. Simplify expression: Step 2: Calculate (f/g)(x). (f/g)(x) = f(x) / g(x) = (2/x) / (x - 3). Simplify to (f/g)(x) = 2 / (x(x - 3)). Determine domain: Step 3: Simplify the expression for (f/g)(x). (f/g)(x) = 2 / (x^2 - 3x). Write domain: Step 4: Determine the domain of (f/g)(x). The denominator x^2 - 3x cannot be zero. Factorize: x(x - 3) = 0. x = 0 or x = 3. Thus, x cannot be 0 or 3. Step 5: Write the domain using interval notation. Domain of (f/g): x ∈ ℝ except x = 0, 3. In interval notation: (-∞, 0) ∪ (0, 3) ∪ (3, ∞).

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Suppose that the functions $f$ and $g$ are defined as follows. $\newline$ $f(x)=\frac{2}{x}\quad g(x)=x-3$ $\newline$ Find $\frac{f}{g}$ . Then, give its domain using an interval or union of intervals Simplify your answers. $\newline$ $\left(\frac{f}{g}\right)(x)=\boxed{\phantom{0}}$ $\newline$ Domain of $\frac{f}{g}$ :

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Math Problems

Algebra 1

Domain and range of square root functions: equations

Full solution

Q. Suppose that the functions

f

and

g

are defined as follows.

\newline

f(x)=\frac{2}{x}\quad g(x)=x-3

\newline

Find

\frac{f}{g}

. Then, give its domain using an interval or union of intervals Simplify your answers.

\newline

\left(\frac{f}{g}\right)(x)=\boxed{\phantom{0}}

\newline

Domain of

\frac{f}{g}

Define functions: question_prompt: Find the function $(f/g)(x)$ and determine its domain.
Calculate $(f/g)(x)$ : Step $1$ : Define the functions $f(x)$ and $g(x)$ . $\newline$ $f(x) = \frac{2}{x}$ , $g(x) = x - 3$ .
Simplify expression: Step $2$ : Calculate $(f/g)(x)$ . $\newline$ $(f/g)(x) = \frac{f(x)}{g(x)} = \frac{2/x}{x - 3}$ . $\newline$ Simplify to $(f/g)(x) = \frac{2}{x(x - 3)}$ .
Determine domain: Step $3$ : Simplify the expression for $(f/g)(x)$ . $\newline$ $(f/g)(x) = \frac{2}{x^2 - 3x}$ .
Write domain: Step $4$ : Determine the domain of $(f/g)(x)$ . $\newline$ The denominator $x^2 - 3x$ cannot be zero. $\newline$ Factorize: $x(x - 3) = 0$ . $\newline$ $x = 0$ or $x = 3$ . $\newline$ Thus, $x$ cannot be $0$ or $3$ .
Write domain: Step $4$ : Determine the domain of $(f/g)(x)$ . $\newline$ The denominator $x^2 - 3x$ cannot be zero. $\newline$ Factorize: $x(x - 3) = 0$ . $\newline$ $x = 0$ or $x = 3$ . $\newline$ Thus, $x$ cannot be $0$ or $3$ . Step $5$ : Write the domain using interval notation. $\newline$ Domain of $(f/g)$ : $x \in \mathbb{R}$ except $x^2 - 3x$ $0$ . $\newline$ In interval notation: $x^2 - 3x$ $1$ .