Domain & Range Of F(x) = |x-3|+6: Explained!

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Hey guys! Let's dive into finding the domain and range of the function f(x) = |x-3| + 6. This is a classic problem in mathematics, and understanding how to solve it will help you tackle many other similar problems. We'll break it down step-by-step, so you can easily grasp the concepts. This article provides the answer to this question: What are the domain and range of f(x)=|x-3|+6 ?

Understanding Domain and Range

Before we jump into the specifics of our function, let's quickly recap what domain and range mean.

  • Domain: The domain of a function is the set of all possible input values (x-values) that the function can accept without causing any mathematical errors (like division by zero or taking the square root of a negative number).
  • Range: The range of a function is the set of all possible output values (y-values or f(x) values) that the function can produce from the valid input values.

Analyzing the Function f(x) = |x-3| + 6

Our function is f(x) = |x-3| + 6. This function involves an absolute value and a constant. Let's analyze each part to determine the domain and range.

Domain Analysis

When determining the domain of f(x) = |x-3| + 6, we need to identify any restrictions on the input values (x-values). Ask yourself, are there any values of x that would make this function undefined? The key part of this function is the absolute value, |x-3|. The absolute value function is defined for all real numbers. You can plug in any real number into |x-3|, and you will always get a valid output. There are no square roots, no fractions with x in the denominator, or any other operations that might restrict the values of x. Because the absolute value |x - 3| is defined for all real numbers, and adding 6 doesn't introduce any new restrictions, the domain of f(x) is all real numbers. This means that x can be any real number, whether it's positive, negative, zero, a fraction, or an irrational number.

So, considering that the absolute value function |x - 3| is defined for all real numbers, and adding 6 doesn't introduce any new restrictions, we can confidently say that the domain of f(x) is all real numbers. Mathematically, we can express this as:

Domain: {x | x is all real numbers}

This notation means “the set of all x such that x is a real number.” It's a concise way to describe that x can be any number on the number line.

In simpler terms, you can plug in any value for x into the function f(x) = |x - 3| + 6, and you will always get a valid result. There are no restrictions or limitations on what x can be, so the domain is all real numbers.

Range Analysis

Now, let's find the range of f(x) = |x-3| + 6. The range consists of all possible output values (y-values or f(x) values) that the function can produce. To determine the range, we need to understand how the absolute value and the constant affect the output.

The absolute value function |x-3| always returns a non-negative value. In other words, |x-3| is always greater than or equal to 0, regardless of the value of x. The minimum value of |x-3| is 0, which occurs when x = 3. The absolute value |x - 3| is always non-negative, meaning it's either zero or positive. Therefore, the smallest value that |x - 3| can take is 0.

Since |x-3| is always greater than or equal to 0, we have:

|x-3| ≥ 0

Now, let's consider the entire function f(x) = |x-3| + 6. We are adding 6 to the absolute value. Since the smallest value of |x-3| is 0, the smallest value of f(x) will be when |x-3| = 0.

f(x) = |x-3| + 6 ≥ 0 + 6 f(x) ≥ 6

This inequality tells us that the function f(x) will always be greater than or equal to 6. In other words, the smallest possible output value (y-value) is 6. Adding 6 to |x - 3| shifts the entire graph up by 6 units. Since the minimum value of |x - 3| is 0, the minimum value of f(x) is 0 + 6 = 6.

Thus, the range of the function f(x) = |x-3| + 6 is all real numbers greater than or equal to 6. We can express this mathematically as:

Range: {y | y ≥ 6}

This notation means “the set of all y such that y is greater than or equal to 6.” It indicates that the output values of the function will always be 6 or higher.

In summary, the absolute value |x - 3| is always non-negative, so the minimum value of |x - 3| + 6 is 6. This means that the range of the function is all y values greater than or equal to 6. The function f(x) = |x - 3| + 6 will never produce a value less than 6.

Conclusion

Alright, let's wrap things up! After our analysis:

  • The domain of the function f(x) = |x-3| + 6 is all real numbers.
  • The range of the function f(x) = |x-3| + 6 is {y | y ≥ 6}.

Therefore, the correct answer is:

A. Domain: x | x is all real numbers} Range {y | y ≥ 6

Understanding domain and range is crucial for analyzing functions. By identifying the possible input and output values, you can gain a deeper insight into the behavior of the function. Keep practicing, and you'll become a pro in no time! Remember, the domain is all possible x values, and the range is all possible y values. For f(x) = |x - 3| + 6, the absolute value doesn't restrict the x values, but it does affect the y values by ensuring they are always greater than or equal to 6. So the domain is all real numbers, and the range is y ≥ 6. Great job, guys! You've successfully navigated through finding the domain and range of this function. Keep up the excellent work, and you'll master these concepts in no time!