Rational Function Analysis: Domain, Intercepts, Range

Alright guys, let's dive into analyzing the rational function $g(x)=\frac{x^2-3x-4}{x^2-4}$. We're going to break down its key characteristics: the domain, x-intercept(s), y-intercept(s), and range. Understanding these features is crucial for sketching the graph and comprehending the behavior of the function. So, buckle up, and let's get started!

1. Domain of the Rational Function

The domain of a rational function is all real numbers except for the values that make the denominator equal to zero. Why? Because division by zero is undefined! In our case, the denominator is $x^2 - 4$. We need to find the values of x that make $x^2 - 4 = 0$.

We can factor the denominator as a difference of squares: $x^2 - 4 = (x - 2)(x + 2)$. Setting each factor to zero gives us $x - 2 = 0$ or $x + 2 = 0$. Solving these equations, we find $x = 2$ and $x = -2$.

Therefore, the domain of $g(x)$ is all real numbers except $x = 2$ and $x = -2$. In set notation, we can write this as $\mathbb{R} \setminus \{-2, 2\}$. This means that the graph of the function will have vertical asymptotes at $x = 2$ and $x = -2$. These asymptotes are like invisible walls that the function approaches but never crosses. Understanding the domain is the foundation for analyzing the rest of the function's characteristics.

So, to reiterate, when you're looking for the domain of a rational function, always focus on the denominator. Find the values that make it zero and exclude them from the set of all real numbers. This simple step is crucial for understanding the behavior and graph of the function. Remember that identifying domain restrictions early will also guide you to determine the location of vertical asymptotes, which are critical for accurate graphing.

2. X-intercept(s) of the Rational Function

The x-intercepts of a function are the points where the graph crosses the x-axis. At these points, the y-value (or the function value) is equal to zero. So, to find the x-intercepts of $g(x)$, we need to solve the equation $g(x) = 0$.

This means we need to find the values of x that make the numerator equal to zero, while ensuring that these values are within the domain of the function. Our numerator is $x^2 - 3x - 4$. Let's factor this quadratic: $x^2 - 3x - 4 = (x - 4)(x + 1)$.

Setting each factor to zero gives us $x - 4 = 0$ or $x + 1 = 0$. Solving these equations, we find $x = 4$ and $x = -1$. Now, we need to check if these values are in the domain of $g(x)$. Since our domain excludes $x = 2$ and $x = -2$, both $x = 4$ and $x = -1$ are valid x-intercepts.

Therefore, the x-intercepts of $g(x)$ are $x = 4$ and $x = -1$. These correspond to the points $(4, 0)$ and $(-1, 0)$ on the graph. The x-intercepts give us key points where the function crosses or touches the x-axis. They, along with the asymptotes, help define the overall shape and behavior of the function.

Remember: always verify that the x-values you find are within the function's domain. If an x-value makes both the numerator and the denominator zero, it indicates a hole in the graph, not necessarily an x-intercept. Determining the x-intercepts is critical for understanding where the function changes its sign. These sign changes frequently happen at the x-intercepts.

3. Y-intercept(s) of the Rational Function

The y-intercept is the point where the graph crosses the y-axis. This occurs when $x = 0$. To find the y-intercept of $g(x)$, we simply need to evaluate $g(0)$.

Substituting $x = 0$ into the function, we get:

$g(0) = \frac{0^2 - 3(0) - 4}{0^2 - 4} = \frac{-4}{-4} = 1$

Therefore, the y-intercept of $g(x)$ is $y = 1$. This corresponds to the point $(0, 1)$ on the graph. The y-intercept provides a quick and easy point to plot, helping to orient the graph on the coordinate plane. This point provides essential information about the function's behavior near the y-axis.

Finding the y-intercept is usually straightforward, just plug in x = 0, but it's always a good practice to confirm that x=0 is in the function’s domain. If x=0 is not in the domain, then there will be no y-intercept.

4. Range of the Rational Function

The range of a function is the set of all possible output values (y-values). Determining the range of a rational function can be tricky and often involves a combination of algebraic techniques and graphical analysis. There isn't a direct algebraic method to find the range in all cases, especially for more complex rational functions.

For our function, $g(x) = \frac{x^2 - 3x - 4}{x^2 - 4}$, we can use the information we've gathered so far (domain, intercepts, asymptotes) to get an idea of the range. We know that there are vertical asymptotes at $x = 2$ and $x = -2$. To get a better idea of the function's behavior near these asymptotes, you could test values slightly to the left and right of each asymptote. Furthermore, we can also find the horizontal asymptote by examining the degrees of the numerator and denominator.

Since the degree of the numerator and the degree of the denominator are the same (both are 2), there exists a horizontal asymptote at y = (leading coefficient of numerator) / (leading coefficient of denominator). Therefore, there is a horizontal asymptote at y = 1/1 = 1.

The horizontal asymptote suggests that as x approaches positive or negative infinity, g(x) will approach 1. However, a horizontal asymptote does not guarantee that the function will never take on the value of y = 1. To determine if it does, we can set g(x) = 1 and solve for x.

$\frac{x^2 - 3x - 4}{x^2 - 4} = 1$ $x^2 - 3x - 4 = x^2 - 4$ $-3x = 0$ x = 0

Since we found a solution x = 0, this means that the function actually does equal 1 when x = 0 (which we already knew from the y-intercept). More detailed analysis, potentially using calculus or graphing software, is usually needed to precisely determine the range. We would need to find local max/min to determine range accurately. Without a graphing utility or further analysis, we cannot determine the range with certainty. We know that the function takes on values near the horizontal asymptote y = 1. It also contains points above and below the x-axis, which indicates the function takes on positive and negative values. In this case, we cannot determine an exact answer for the range without more advanced methods. In general, the range is difficult to determine analytically and is better approximated by plotting the graph.

In Summary:

• Domain: All real numbers except $x = -2$ and $x = 2$ (or $x \neq -2, 2$)
• X-intercepts: $x = 4$ and $x = -1$
• Y-intercept: $y = 1$
• Range: Can not be determined without more advanced methods.

By analyzing these characteristics, we gain a solid understanding of the behavior of the rational function $g(x)$. Remember, understanding the domain is the first step, followed by finding intercepts and asymptotes. These elements work together to paint a complete picture of the function's graph.