Question

find the domain of $f(x) = \frac{8x}{x - 2}$\
\
\bigcirc all real numbers\
\bigcirc all real numbers except 2\
\bigcirc all real numbers except 0\
\bigcirc all real numbers except 0 and 2\
\
question 8\
divide and simplify: $\frac{a^2 - 9}{a^2 - 49} \div \frac{a + 3}{a + 7}$\
\
\bigcirc $\frac{a + 3}{a + 7}$\
\bigcirc $\frac{9(a + 3)}{49(a - 7)}$\
\bigcirc $\frac{a - 3}{(a - 7)^2}$\
\bigcirc $\frac{a - 3}{a - 7}$

Explanation:

First Problem: Find the domain of $f(x)=\frac{8x}{x-2}$

Step1: Identify undefined condition

A rational function is undefined when its denominator equals 0.
Set denominator to 0: $x-2=0$

Step2: Solve for x

Solve the equation: $x=2$

Step3: Define valid domain

All real numbers except the value that makes the function undefined.

Step1: Rewrite division as multiplication

Dividing by a fraction is multiplying by its reciprocal.
$\frac{a^2-9}{a^2-49} \times \frac{a+7}{a+3}$

Step2: Factor all polynomials

Use difference of squares: $a^2-b^2=(a-b)(a+b)$
$\frac{(a-3)(a+3)}{(a-7)(a+7)} \times \frac{a+7}{a+3}$

Step3: Cancel common factors

Cancel $(a+3)$ and $(a+7)$ from numerator and denominator.
$\frac{a-3}{a-7}$

Answer:

all real numbers except 2

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Second Problem: Divide and simplify $\frac{a^2-9}{a^2-49} \div \frac{a+3}{a+7}$