Problem 32 - Precalculus: Functions and Graphs

Question

Sketch the graph of \(f\) $$f(x)=\frac{x^{2}-4}{x^{2}+1}$$

Step-by-Step Solution

Answer

The graph of \( f(x) = \frac{x^2 - 4}{x^2 + 1} \) is symmetric about the y-axis with x-intercepts at \(x = 2\), \(x = -2\), a y-intercept at \(y = -4\), and a horizontal asymptote at \(y = 1\).

Step 1: Identify the Domain

First, let's identify the domain of the function. The function is defined for all real numbers because the denominator, \(x^2 + 1\), is never zero. Therefore, the domain of \(f(x)\) is all real numbers, \(x \in \mathbb{R}\).

Step 2: Find the Intercepts

To find the x-intercepts of the function, set \(f(x) = 0\). This occurs when the numerator is zero: \(x^2 - 4 = 0\). Solving \(x^2 - 4 = 0\) gives \(x = 2\) and \(x = -2\). Therefore, the x-intercepts are \((2, 0)\) and \((-2, 0)\). The y-intercept is found by evaluating \(f(0)\): \(f(x) = \frac{x^2 - 4}{x^2 + 1}\) gives \(f(0) = \frac{-4}{1} = -4\). Hence, the y-intercept is \((0, -4)\).

Step 3: Determine Symmetry

Check for symmetry by evaluating \(f(-x)\): \(f(-x) = \frac{(-x)^2 - 4}{(-x)^2 + 1} = \frac{x^2 - 4}{x^2 + 1} = f(x)\). This means \(f(x)\) is an even function, and its graph is symmetric about the y-axis.

Step 4: Analyze Asymptotic Behavior

Determine horizontal asymptotes by evaluating the end behavior of \(f(x)\) as \(x \to \infty\) or \(x \to -\infty\). As \(x\) becomes very large, both the numerator and the denominator are dominated by \(x^2\), so \(f(x) = \frac{x^2 - 4}{x^2 + 1} \approx \frac{x^2}{x^2} = 1\). Therefore, there is a horizontal asymptote at \(y = 1\). There are no vertical asymptotes because the denominator \(x^2 + 1\) is never zero.

Step 5: Plot Key Points and Asymptotes

Using the information gathered, plot the intercepts: \((2, 0)\), \((-2, 0)\), and \((0, -4)\). Add the horizontal asymptote at \(y = 1\). Note the symmetry and sketch the curve approaching the asymptote as \(|x|\) becomes large while crossing through the intercepts.

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (usually represented by \(x\)) that allow the function to produce real and defined outputs. In the function \(f(x)=\frac{x^{2}-4}{x^{2}+1}\), we observe the denominator, \(x^2+1\), can never equal zero. Since there's no zero in the denominator, \(f(x)\) is defined for every real number. Therefore, the domain of this function is the entire set of real numbers, \(x \in \mathbb{R}\). This means for any real number you choose to plug into \(f(x)\), it will yield a real output value. Understanding the domain is crucial because it tells us which values of \(x\) can be used without encountering any undefined mathematical operations like division by zero or the square root of a negative number.

Intercepts of a Function

Intercepts are points where the graph of a function crosses the x-axis and y-axis. To find the x-intercepts, we set \(f(x) = 0\). For \(f(x)=\frac{x^{2}-4}{x^{2}+1}\), the numerator must be zero, while the denominator shouldn't be zero (though in this case, it isn't). Solving \(x^2 - 4 = 0\) gives us x-intercepts at \((2, 0)\) and \((-2, 0)\). These points are where the graph meets the x-axis.

For the y-intercept, we evaluate \(f(0)\). Substituting \(x = 0\) into the function yields \(f(0) = \frac{-4}{1} = -4\). Therefore, the y-intercept is at \((0, -4)\). Knowing intercepts helps to quickly identify key points on the graph and its basic alignment with the axes.

Symmetry of Functions

Symmetry in functions can be incredibly useful for graphing, as it tells us about the balance of the graph around specific axes. A function can be symmetric with respect to the y-axis, the x-axis, or the origin. For our function \(f(x)=\frac{x^{2}-4}{x^{2}+1}\), checking for symmetry involves substituting \(-x\) for \(x\) in the equation to check if \(f(-x) = f(x)\).

After evaluating, we find \(f(-x) = \frac{x^2 - 4}{x^2 + 1} = f(x)\), confirming the function is even and symmetric about the y-axis. This property implies if you folded the graph along the y-axis, both halves would match perfectly. This concept simplifies plotting since knowing one side of the graph provides complete information about the other side.

Asymptotic Behavior

Asymptotic behavior gives insights about how a function behaves as \(x\) approaches extremely large positive or negative values. For the function \(f(x)=\frac{x^{2}-4}{x^{2}+1}\), we study its end behavior to find horizontal asymptotes. As \(x\) approaches infinity or negative infinity, the terms \(x^2\) in both the numerator and denominator dominate, leading to the simplified form \(\frac{x^2}{x^2}\), which equals 1.

This tells us the graph has a horizontal asymptote at \(y = 1\). No matter how large \(x\) gets, the function approaches the line \(y = 1\) but never actually reaches it. Additionally, vertical asymptotes are absent because \(x^2 + 1\) cannot be zero. Asymptotic analysis helps anticipate the graph's behavior at its extremes, aiding in more accurate graphing.