Questions: Quiz 1 Find the domain of the function. g(x)=3x/(x^2-25) A. x x>25 B. x x ≠ -5,5 C. all real numbers D. x x ≠ 0

Quiz 1

Find the domain of the function.
g(x)=3x/(x^2-25)
A. x  x>25
B. x  x ≠ -5,5
C. all real numbers
D. x  x ≠ 0
Transcript text: Quiz 1 Find the domain of the function. \[ g(x)=\frac{3 x}{x^{2}-25} \] A. $\{x \mid x>25\}$ B. $\{x \mid x \neq-5,5\}$ C. all real numbers D. $\{x \mid x \neq 0\}$
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Solution

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Solution Steps

To find the domain of the function g(x)=3xx225 g(x) = \frac{3x}{x^2 - 25} , we need to determine the values of x x for which the function is defined. The function is undefined where the denominator is zero. Therefore, we solve the equation x225=0 x^2 - 25 = 0 to find the values of x x that make the denominator zero. These values will be excluded from the domain.

Step 1: Identify the Function's Denominator

The function given is g(x)=3xx225 g(x) = \frac{3x}{x^2 - 25} . To find the domain, we need to identify where the function is undefined. This occurs when the denominator is zero.

Step 2: Solve for Denominator Equal to Zero

Set the denominator equal to zero and solve for x x : x225=0 x^2 - 25 = 0 This can be factored as: (x5)(x+5)=0 (x - 5)(x + 5) = 0 Solving these equations gives: x=5andx=5 x = 5 \quad \text{and} \quad x = -5

Step 3: Determine the Domain

The function is undefined at x=5 x = 5 and x=5 x = -5 . Therefore, the domain of the function is all real numbers except x=5 x = 5 and x=5 x = -5 .

Final Answer

{xx5,5}\boxed{\{x \mid x \neq -5, 5\}}

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