Domain And Range Of F(x)=(x-6)/(x^2-3x-18) A Step-by-Step Guide

by ADMIN 64 views

Determining the domain and range of functions can be a tricky task, especially when dealing with rational functions. Rational functions, which are essentially fractions with polynomials in the numerator and denominator, present unique challenges due to potential restrictions on the input values and the resulting output values. In this comprehensive guide, we will walk you through a step-by-step process to find the domain and range of the function $f(x) = \frac{x-6}{x^2 - 3x - 18}$. We'll break down each step, providing clear explanations and examples to ensure you grasp the underlying concepts.

Understanding Domain and Range

Before diving into the specifics of our function, let's first clarify what domain and range mean in the context of functions. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's the collection of all numbers you can plug into the function without causing any mathematical errors, such as division by zero or taking the square root of a negative number. Think of the domain as the function's playground – the set of inputs where it can happily play without encountering any obstacles.

On the other hand, the range of a function is the set of all possible output values (y-values or f(x)-values) that the function can produce when you plug in values from its domain. It's the collection of all the results you can get out of the function. Imagine the range as the function's trophy case – it holds all the possible awards (outputs) the function can achieve when given valid inputs. Finding the range can sometimes be more challenging than finding the domain, as it requires analyzing the function's behavior and identifying any limitations on its output values.

For rational functions, the domain is primarily restricted by values that make the denominator zero, as division by zero is undefined in mathematics. The range can be affected by horizontal asymptotes, which the function approaches but never quite reaches, and by any other limitations imposed by the function's structure. Now, let's apply these concepts to our specific function and unravel its domain and range.

Step 1: Finding the Domain

To determine the domain of the function $f(x) = \frac{x-6}{x^2 - 3x - 18}$, our primary focus is on the denominator. As we discussed earlier, the domain consists of all real numbers except those that make the denominator equal to zero. Division by zero is undefined, so we must identify and exclude these values from our domain.

Our denominator is the quadratic expression $x^2 - 3x - 18$. To find the values of x that make this expression zero, we need to solve the equation $x^2 - 3x - 18 = 0$. This is a quadratic equation, and we can solve it by factoring, using the quadratic formula, or completing the square. In this case, factoring is the most straightforward approach.

We need to find two numbers that multiply to -18 and add up to -3. These numbers are -6 and +3. Therefore, we can factor the quadratic as follows:

x2−3x−18=(x−6)(x+3)x^2 - 3x - 18 = (x - 6)(x + 3)

Now, we set each factor equal to zero and solve for x:

  • x−6=0⇒x=6x - 6 = 0 \Rightarrow x = 6

  • x+3=0⇒x=−3x + 3 = 0 \Rightarrow x = -3

These are the values of x that make the denominator zero. Consequently, they must be excluded from the domain. In interval notation, we can express the domain as the set of all real numbers except 6 and -3. This can be written as:

Domain: $(-\infty, -3) \cup (-3, 6) \cup (6, \infty)$

This notation indicates that the domain includes all real numbers less than -3, all real numbers between -3 and 6, and all real numbers greater than 6. The union symbol (∪\cup) combines these intervals into a single set. By excluding -3 and 6, we ensure that we are only considering input values for which the function is properly defined. Now that we've successfully determined the domain, let's move on to the more challenging task of finding the range.

Step 2: Simplifying the Function

Before we tackle the range of the function, it's often helpful to simplify the function as much as possible. This can reveal hidden properties and make it easier to analyze the function's behavior. Simplification involves canceling out common factors in the numerator and denominator, if any exist. In our case, we have:

f(x)=x−6x2−3x−18f(x) = \frac{x-6}{x^2 - 3x - 18}

We already factored the denominator in the previous step as $(x - 6)(x + 3)$. So, we can rewrite the function as:

f(x)=x−6(x−6)(x+3)f(x) = \frac{x-6}{(x - 6)(x + 3)}

Notice that we have a common factor of $(x - 6)$ in both the numerator and the denominator. We can cancel this factor out, but with a crucial caveat: we must remember that x cannot be equal to 6, as this would make the original denominator zero. With this in mind, we can simplify the function as follows:

f(x) = \frac{1}{x + 3}$, $x \neq 6

This simplified form is much easier to work with. However, we must not forget the restriction $x \neq 6$. This restriction will play a vital role when we determine the range of the function. Simplifying the function has unveiled a crucial aspect of its behavior: it now resembles a basic reciprocal function, but with a horizontal shift and a hole at $x = 6$. This hole represents a point discontinuity in the graph of the function, and it will affect the range.

Step 3: Identifying Horizontal Asymptotes

To understand the range of the function, a critical step is to identify any horizontal asymptotes. A horizontal asymptote is a horizontal line that the graph of the function approaches as x approaches positive or negative infinity. In simpler terms, it's a line that the function gets arbitrarily close to but never actually crosses (unless there are other factors at play, such as the hole we identified earlier).

For rational functions, horizontal asymptotes are determined by comparing the degrees of the polynomials in the numerator and denominator. The degree of a polynomial is the highest power of x that appears in the expression. In our simplified function, $f(x) = \frac{1}{x + 3}$, the degree of the numerator (1) is 0, as it's a constant. The degree of the denominator ($x + 3$) is 1, as the highest power of x is 1.

There's a rule that governs the existence of horizontal asymptotes based on these degrees:

  • If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y = 0 (the x-axis).
  • If the degree of the numerator is equal to the degree of the denominator, there is a horizontal asymptote at y = the ratio of the leading coefficients (the coefficients of the terms with the highest powers of x).
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (but there may be a slant asymptote).

In our case, the degree of the numerator (0) is less than the degree of the denominator (1). Therefore, we have a horizontal asymptote at y = 0. This means that as x becomes very large (positive or negative), the function's values will approach 0, but they will never actually reach 0, except possibly at the hole we identified earlier. The horizontal asymptote gives us a crucial piece of information about the range: the function will not take on the value 0. However, we still need to consider the impact of the hole on the range.

Step 4: Considering the Hole

As we simplified the function, we canceled out the factor $(x - 6)$, which introduced a hole in the graph at $x = 6$. This hole is a point discontinuity, meaning that the function is not defined at that specific x-value. The presence of a hole can affect the range of the function, as it represents a missing y-value. To find the y-coordinate of the hole, we plug the x-value (6) into the simplified function:

f(6)=16+3=19f(6) = \frac{1}{6 + 3} = \frac{1}{9}

This tells us that there is a hole in the graph at the point $(6, \frac1}{9})$. This means that the function will take on all y-values except $\frac{1}{9}$. We already knew that the function would not take on the value 0 due to the horizontal asymptote. Now, we have another value to exclude from the range $\frac{1{9}$. The hole acts like a gap in the range, preventing the function from achieving that specific output value.

Step 5: Determining the Range

Now that we've identified the horizontal asymptote and considered the hole, we have all the pieces we need to determine the range of the function. We know that the function approaches the horizontal asymptote at y = 0 but never actually reaches it. We also know that there's a hole in the graph at $y = \frac{1}{9}$, meaning that this value is also excluded from the range.

The simplified function, $f(x) = \frac{1}{x + 3}$, can take on any y-value except 0 and $\frac{1}{9}$. In interval notation, we can express the range as follows:

Range: $(-\infty, 0) \cup (0, \frac{1}{9}) \cup (\frac{1}{9}, \infty)$

This notation indicates that the range includes all real numbers less than 0, all real numbers between 0 and $\frac{1}{9}$, and all real numbers greater than $\frac{1}{9}$. The union symbols (∪\cup) combine these intervals into a single set. By excluding 0 and $\frac{1}{9}$, we accurately represent the possible output values of the function. We've successfully navigated the complexities of finding the range, taking into account the horizontal asymptote and the hole.

Summary: Domain and Range of $f(x) = \frac{x-6}{x^2 - 3x - 18}$

Let's recap our findings. For the function $f(x) = \frac{x-6}{x^2 - 3x - 18}$, we determined the following:

  • Domain: $(-\infty, -3) \cup (-3, 6) \cup (6, \infty)$
  • Range: $(-\infty, 0) \cup (0, \frac{1}{9}) \cup (\frac{1}{9}, \infty)$

We found the domain by identifying the values of x that make the denominator zero and excluding them. We found the range by simplifying the function, identifying the horizontal asymptote, and considering the hole in the graph. This step-by-step approach can be applied to other rational functions as well.

Conclusion

Finding the domain and range of rational functions requires a careful analysis of the function's structure and behavior. By systematically identifying restrictions on the input values (domain) and the resulting output values (range), we can gain a deep understanding of the function's properties. This comprehensive guide has equipped you with the tools and knowledge to tackle the domain and range of rational functions with confidence. Remember to always consider the denominator, simplify the function, identify horizontal asymptotes, and account for any holes in the graph. With practice and a solid understanding of these concepts, you'll be able to master the art of domain and range determination.