Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The graph of \(f(x) = \frac{2}{(x+1)^2}\) has a vertical asymptote at \(x = -1\), a horizontal asymptote at \(y = 0\), and resembles a parabola facing upwards on either side of the asymptote.

Step 1: Determine the Domain

To find the domain of the function, we need to identify any values of \(x\) that make the denominator zero. For the function \(f(x) = \frac{2}{(x+1)^2}\), the denominator is \((x+1)^2\), which is zero when \((x+1)^2 = 0\), meaning \(x+1 = 0\) or \(x = -1\). Therefore, the domain of \(f\) is all real numbers except \(x = -1\).

Step 2: Identify Vertical Asymptote

From the domain analysis, we know that the denominator becomes zero at \(x = -1\). This means that there is a vertical asymptote at \(x = -1\). The graph approaches this line but never touches or crosses it.

Step 3: Determine Horizontal Asymptote

As \(x\) approaches infinity, the value of \(f(x) = \frac{2}{(x+1)^2}\) approaches zero because the denominator grows much larger than the numerator. Therefore, there is a horizontal asymptote at \(y = 0\).

Step 4: Analyze Symmetry

Since \(f(-x) = \frac{2}{((-x)+1)^2} = \frac{2}{(1-x)^2}\) is not equal to \(f(x)\), the function is not symmetric about the y-axis. Furthermore, since \(f(-x)\) is not equal to \(-f(x)\), the function is not symmetric about the origin. There is no symmetry.

Step 5: Evaluate Limits at Asymptotes

Calculate the behavior of the function as \(x\) approaches \(-1\) from the left and right.\[ \lim_{{x \to -1^-}} f(x) = \infty \] \[ \lim_{{x \to -1^+}} f(x) = \infty \] Since both limits go to infinity, the graph rises to infinity on both sides of the vertical asymptote.

Step 6: Sketch the Graph

Using the information gathered: The graph has a vertical asymptote at \(x = -1\) and a horizontal asymptote at \(y = 0\). Evaluate some values to get points. - For \(x = 0\), \(f(0) = \frac{2}{1^2} = 2\) - For \(x = 1\), \(f(1) = \frac{2}{4} = 0.5\)- For \(x = -2\), \(f(-2) = 2\) Plot these and shape the graph as described: As \(x\) approaches \(-1\), the graph ascends exceedingly high, and as \(x\) goes towards infinity, it follows the horizontal line \(y = 0\).

Key Concepts

Domain of a Function

The domain of a function is the complete set of values that can be input into the function without causing any mathematical mishaps, like division by zero. For functions like the one given, which is a rational function, we focus on the denominator.
In the function \(f(x) = \frac{2}{(x+1)^2}\), the denominator \((x+1)^2\) is zero when \(x = -1\). Division by zero is undefined, so \(x = -1\) must be excluded from the domain.
Thus, the domain of \(f\) includes all real numbers except \(x = -1\). Pay attention to rational functions, as identifying points where the denominator is zero helps determine their domain.

Asymptotes

Asymptotes are invisible lines that a graph approaches but never actually reaches. They're either vertical, horizontal, or slant and help understand the long-term behavior of a function.

• Vertical Asymptotes: Occur where the function's denominator is zero. For \(f(x) = \frac{2}{(x+1)^2}\), this happens at \(x = -1\). The graph shoots up to infinity as it approaches this line.
• Horizontal Asymptotes: Describe the behavior as \(x\) goes to positive or negative infinity. Here, \(f(x)\) approaches 0 as \(x\) grows because the denominator becomes much larger than the numerator. Therefore, there's a horizontal asymptote at \(y = 0\).

Remember, vertical asymptotes indicate non-permissible \(x\)-values, while horizontal asymptotes indicate the value \(f(x)\) settles to over the long haul.

Symmetry in Graphs

Examining symmetry can simplify graph analysis by identifying repetitive patterns. Symmetry can be with respect to the y-axis or the origin.

• Y-axis Symmetry: Occurs if \(f(-x) = f(x)\). This implies that the left and right hand sides of the graph mirror each other. In this case, \(f(-x) eq f(x)\), meaning this symmetry isn't present.
• Origin Symmetry: Occurs if \(f(-x) = -f(x)\). This means the graph is mirror-symmetric about the origin. Again, for this function, \(f(-x) eq -f(x)\), indicating no origin symmetry.

Knowing these forms of symmetry helps predict the overall graph shape and behavior, but here, the function lacks any symmetry.

Limits and Continuity

Limits explore a function's behavior as it approaches specific points or infinity. Continuity involves the smoothness of the function's graph.

• Limits at Asymptotes: For vertical asymptotes, check limits on both sides. As \(x\) approaches \(-1\) from either direction, \(f(x)\) heads towards infinity: \[ \lim_{{x \to -1^-}} f(x) = \infty\] \[ \lim_{{x \to -1^+}} f(x) = \infty\] Thus, there's a break in the graph at \(x = -1\).
• Behavior at Infinity: To gauge horizontal asymptotes, consider \(x\) approaching infinity. For \(f(x)\), this limit is 0, highlighting a horizontal asymptote at \(y = 0\).

A function is continuous when it has no abrupt breaks. While \(f(x)\) has a vertical asymptote, it remains smooth elsewhere, barring this point.