Analyzing the Domain of 5\(5x+2.5): Avoiding Undefined Points

Question

Look at the following function:

55x+212 \frac{5}{5x+2\frac{1}{2}}

What is the domain of the function?

Video Solution

Solution Steps

00:07 First, let's ask: Does this function have a domain? If yes, what is it?
00:13 To find the domain, remember; We can't divide by zero. That's important!
00:18 So, let's find out: What value makes the denominator zero?
00:23 Now, let's isolate X. This helps us see the problem clearly.
00:51 Next step: Multiply by the reciprocal. It's like flipping fractions!
00:58 Remember: Multiply numerator by numerator, and denominator by denominator.
01:05 Let's simplify this. We want it to look as simple as possible!
01:09 And there you have it! That's the solution to our question.

Step-by-Step Solution

The given function is:

55x+212 \frac{5}{5x + 2\frac{1}{2}}

To determine the domain, we ensure that the denominator is not equal to zero because division by zero is undefined. So, we start by evaluating 5x+2120 5x + 2\frac{1}{2} \neq 0 .
This requires conversion of the mixed number 212 2\frac{1}{2} into an improper fraction or decimal:

212=2+12=42+12=52 2\frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}

Substituting the improper fraction, our equation becomes:

5x+520 5x + \frac{5}{2} \neq 0

To clear the fraction, multiply through by 2, yielding:

25x+50 2 \cdot 5x + 5 \neq 0

10x+50 10x + 5 \neq 0

Solving for x x by subtracting 5 from both sides informs us:

10x5 10x \neq -5

Now, divide by 10:

x510 x \neq -\frac{5}{10}

Simplify the fraction:

x12 x \neq -\frac{1}{2}

This solution directly identifies the x x value not in the domain:

The domain of the function is all real numbers except x=12 x = -\frac{1}{2} .

Therefore, the domain of the function is x12\mathbf{x \ne -\frac{1}{2}}.

Hence, the correct choice from the given options is:

Choice 4: x12 x \ne -\frac{1}{2}

Answer

x12 x\ne-\frac{1}{2}