Domain & Range: Find F(x) Where X≠3 And Y≠2

Hey guys! Today, we're diving into a fun little math problem that involves figuring out which function has a specific domain and range. Specifically, we want to find the function where x cannot be 3 (that's our domain) and y cannot be 2 (that's our range). This type of problem often pops up and knowing how to tackle it is super useful. Let's break down each option and see what we find!

Understanding Domains and Ranges

Before we jump into the options, let's quickly recap what domain and range actually mean.

• Domain: The domain of a function is all the possible input values (x-values) that the function can accept without causing any mathematical errors. Think of it as the set of all allowable x values you can plug into the function.
• Range: The range of a function is all the possible output values (y-values) that the function can produce. It’s the set of all possible results you can get after plugging in all the valid x values from the domain.

In our case, we're looking for a function where x can be anything except 3, and y can be anything except 2. This usually indicates a rational function (a fraction where both the numerator and denominator are polynomials) because these types of functions can have restrictions on their domain (values that make the denominator zero) and can sometimes have horizontal asymptotes that restrict the range. When dealing with domains, keep an eye out for scenarios where the denominator of a fraction becomes zero or when taking the square root of a negative number, since those operations are undefined in real numbers.

Analyzing the Options

Okay, let's analyze each option to see if it fits our criteria. Remember, we're looking for a function with a domain of all real numbers except 3, and a range of all real numbers except 2.

Option A: $f(x)=\frac{(x-5)}{(x+3)}$

First, let's find the domain. The denominator cannot be zero, so we solve for when x + 3 = 0, which gives us x = -3. So, the domain is all real numbers except -3. This doesn't match our required domain of all real numbers except 3, so we can eliminate this option right away. To find the range, we consider the horizontal asymptote, which occurs when x approaches very large or very small values. In this case, as x goes to infinity, f(x) approaches 1 (the ratio of the leading coefficients). So, the range is all real numbers except 1. Thus, Option A does not satisfy the given conditions for either domain, or range. Therefore, this option is incorrect.

Option B: $f(x)=\frac{2(x+5)}{(x+3)}$

Again, let's start with the domain. The denominator is x + 3, so the domain is all real numbers except -3. Just like option A, this doesn't match our required domain of all real numbers except 3. To determine the range, we examine the horizontal asymptote. As x approaches infinity, f(x) approaches 2 (the ratio of the leading coefficients, which is 2/1). This means the range is all real numbers except 2, which does match our range requirement. However, since the domain does not match, this option is also incorrect. This option highlights the importance of checking both domain and range, as one correct value doesn't make the entire function correct.

Option C: $f(x)=\frac{2(x+5)}{(x-3)}$

Let's check the domain. The denominator is x - 3, so the domain is all real numbers except 3. This does match our required domain! Now, let's find the range. As x approaches infinity, f(x) approaches 2 (the ratio of the leading coefficients, 2/1). So, the range is all real numbers except 2. This also matches our range requirement! Therefore, option C satisfies both the domain and range conditions.

Option D: $f(x)=\frac{(x+5)}{(x-3)}$

Let's analyze the domain first. The denominator is x - 3, so the domain is all real numbers except 3. This matches the domain requirement we're looking for. Next, let's find the range. As x approaches infinity, f(x) approaches 1 (the ratio of the leading coefficients, 1/1). So, the range is all real numbers except 1. This doesn't match our range requirement of all real numbers except 2. Even though the domain is correct, the range is not, so this option is incorrect.

The Answer

After analyzing all the options, we found that only option C, $f(x)=\frac{2(x+5)}{(x-3)}$, satisfies both the domain requirement (x ≠ 3) and the range requirement (y ≠ 2). Therefore, option C is the correct answer. When trying to determine the domain, you will need to check that the denominator doesn't equal zero. Then when trying to determine the range, you need to find any horizontal asymptotes.

Key Takeaways

• Always remember to check both the domain and the range when solving these types of problems. One correct value does not make the entire function correct.
• Rational functions are your friends (and sometimes your foes!) when dealing with domain and range restrictions.
• Pay close attention to the values that x and y cannot be, as these are your key clues.
• Don't be afraid to eliminate options that don't fit the criteria early on – it saves time!

And there you have it, guys! Hope this breakdown helps you nail similar problems in the future. Keep practicing, and you'll become domain and range masters in no time! Understanding the properties of functions, especially rational functions, is important for solving problems related to domain and range. Remember to consider restrictions such as division by zero and asymptotes, which often dictate the values that x and y cannot take.