Domain Of Functions: Step-by-Step Guide

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Hey guys! Let's dive into the exciting world of functions and, more specifically, how to find their domains. If you've ever wondered what the domain of a function actually is and how to figure it out, you're in the right place. We'll break it down nice and easy, with examples that'll make you a domain-finding pro in no time. Think of the domain as the function's playground – it's all the possible x-values that you can plug into the function without causing any trouble. Let's get started!

What is the Domain of a Function?

So, what exactly is the domain? In simple terms, the domain of a function is the set of all possible input values (usually x-values) that will produce a valid output. Think of it like this: a function is a machine, you feed it an x-value, and it spits out a y-value. The domain is all the x-values you can feed into the machine without breaking it.

There are a few things that can cause problems and restrict the domain:

  • Division by zero: We can't divide by zero, it's a big no-no in math. If our function has a fraction with x in the denominator, we need to make sure the denominator never equals zero.
  • Square roots of negative numbers: In the world of real numbers, we can't take the square root of a negative number. If our function has a square root, the expression inside the square root must be greater than or equal to zero.
  • Logarithms of non-positive numbers: Logarithms are only defined for positive numbers. If our function has a logarithm, the argument (the thing inside the logarithm) must be greater than zero.

Understanding these restrictions is key to finding the domain of any function. Once you know what to look for, it becomes a puzzle-solving adventure!

Example 1: Finding the Domain of y = 4 / (1 - 4x)

Let's start with a classic example:

a) y = 4 / (1 - 4x)

This function has a fraction, which means we need to watch out for division by zero. Our mission is to find the x-values that make the denominator (1 - 4x) equal to zero.

Here's how we do it:

  1. Set the denominator equal to zero: 1 - 4x = 0
  2. Solve for x:
    • Subtract 1 from both sides: -4x = -1
    • Divide both sides by -4: x = 1/4

So, when x is 1/4, the denominator is zero, and our function is undefined. This means 1/4 is not in the domain.

To express the domain, we can say that it's all real numbers except 1/4. We can write this in a few ways:

  • Set notation: {x | x ∈ ℝ, x ≠ 1/4} (This means "the set of all x such that x is a real number and x is not equal to 1/4")
  • Interval notation: (-∞, 1/4) ∪ (1/4, ∞) (This means all numbers from negative infinity up to 1/4, not including 1/4, and all numbers from 1/4 to infinity.)

Therefore, the domain of y = 4 / (1 - 4x) is all real numbers except x = 1/4. See? Not so scary when you break it down!

Example 2: Finding the Domain of y = √(2x + 10) + √(6 - 3x)

Next up, we have a function with square roots:

b) y = √(2x + 10) + √(6 - 3x)

Remember, we can't take the square root of a negative number (in the real number system, anyway!). So, the expressions inside the square roots must be greater than or equal to zero. This gives us two inequalities to solve:

  1. 2x + 10 ≥ 0
  2. 6 - 3x ≥ 0

Let's tackle them one at a time:

  • Solving 2x + 10 ≥ 0:
    • Subtract 10 from both sides: 2x ≥ -10
    • Divide both sides by 2: x ≥ -5
  • Solving 6 - 3x ≥ 0:
    • Subtract 6 from both sides: -3x ≥ -6
    • Divide both sides by -3 (and remember to flip the inequality sign since we're dividing by a negative number!): x ≤ 2

So, we have two conditions: x must be greater than or equal to -5, and x must be less than or equal to 2. To find the domain, we need to find the overlap of these two conditions.

Think of a number line. We have a point at -5 and a point at 2. x needs to be to the right of -5 (including -5) and to the left of 2 (including 2). This means the domain is the interval between -5 and 2, inclusive.

  • Interval notation: [-5, 2]

Therefore, the domain of y = √(2x + 10) + √(6 - 3x) is [-5, 2]. This means the function is only defined for x-values between -5 and 2, including -5 and 2.

Example 3: Finding the Domain of y = 2 / √(3x² + 7x - 6)

Okay, let's crank up the difficulty a little bit with this example:

c) y = 2 / √(3x² + 7x - 6)

This function combines both a fraction and a square root! This means we have two things to worry about:

  1. The expression inside the square root (3x² + 7x - 6) must be greater than or equal to zero (because of the square root).
  2. The expression inside the square root cannot be equal to zero (because it's in the denominator).

Combining these two conditions, we can say that the expression inside the square root must be strictly greater than zero.

So, we need to solve the inequality:

3x² + 7x - 6 > 0

To solve this quadratic inequality, we'll follow these steps:

  1. Factor the quadratic: (3x - 2)(x + 3) > 0

  2. Find the roots (where the expression equals zero):

    • 3x - 2 = 0 => x = 2/3
    • x + 3 = 0 => x = -3

  3. Create a sign chart: We'll use the roots (-3 and 2/3) to divide the number line into three intervals: (-∞, -3), (-3, 2/3), and (2/3, ∞). We'll pick a test value from each interval and plug it into the factored inequality to see if it's positive or negative.

    Interval Test Value (3x - 2) (x + 3) (3x - 2)(x + 3) > 0?
    (-∞, -3) x = -4 - - + Yes
    (-3, 2/3) x = 0 - + - No
    (2/3, ∞) x = 1 + + + Yes

  4. Identify the intervals where the expression is greater than zero: From the sign chart, we see that the expression is greater than zero in the intervals (-∞, -3) and (2/3, ∞).

  • Interval notation: (-∞, -3) ∪ (2/3, ∞)

Therefore, the domain of y = 2 / √(3x² + 7x - 6) is (-∞, -3) ∪ (2/3, ∞). This means the function is defined for all x-values less than -3 and all x-values greater than 2/3.

Key Takeaways for Finding Domains

Okay, guys, we've covered a lot! Let's recap the key takeaways for finding the domain of a function:

  • Identify potential restrictions: Look for fractions (division by zero), square roots (negative numbers inside), and logarithms (non-positive numbers inside).
  • Set up inequalities: For square roots, set the expression inside greater than or equal to zero. For denominators, set the denominator not equal to zero. For logarithms, set the argument greater than zero.
  • Solve the inequalities: Use your algebra skills to solve for x.
  • Express the domain: Use set notation or interval notation to clearly communicate the set of all valid x-values.
  • Use a sign chart for quadratic inequalities: This is a super helpful tool for solving inequalities involving quadratic expressions.

Finding the domain of a function might seem tricky at first, but with practice, you'll get the hang of it. Remember to break down the problem, identify the restrictions, and solve carefully. You've got this! Keep practicing, and you'll be a domain-finding master in no time.

If you found this guide helpful, share it with your friends, and let's conquer the world of math together!