Questions: What is the domain of f(x) = (x-5)/(x-9) and where is f continuous?

Transcript text: What is the domain of $f(x)=\frac{x-5}{x-9}$ and where is $f$ continuous?

Solution

Solution Steps

Step 1: Identify the denominator and find where it is zero

The function $ f(x) = \frac{x-5}{x-9} $ has a denominator $ x-9 $. To find the domain, we need to determine where the denominator is zero, as division by zero is undefined.

Set the denominator equal to zero: \[ x - 9 = 0 \] Solve for $ x $: \[ x = 9 \]

Step 2: Determine the domain

The function $ f(x) $ is defined for all real numbers except where the denominator is zero. From Step 1, we know the denominator is zero at $ x = 9 $. Therefore, the domain of $ f(x) $ is all real numbers except $ x = 9 $.

In interval notation, the domain is: \[ (-\infty, 9) \cup (9, \infty) \]

Step 3: Determine where $ f $ is continuous

A rational function is continuous everywhere except where the denominator is zero. Since the denominator is zero at $ x = 9 $, the function $ f(x) $ is continuous for all real numbers except $ x = 9 $.

In interval notation, $ f(x) $ is continuous on: \[ (-\infty, 9) \cup (9, \infty) \]

Final Answer

The domain of $ f(x) $ is $ \boxed{(-\infty, 9) \cup (9, \infty)} $ and $ f $ is continuous on $ \boxed{(-\infty, 9) \cup (9, \infty)} $.

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