Problem 49 - Algebra 2

Question

Compare each pair of graphs and find any points of intersection. \(y=\frac{1}{x}\) and \(y=\frac{1}{x^{2}}\)

Step-by-Step Solution

Answer

The graphs of the functions \(y=\frac{1}{x}\) and \(y=\frac{1}{x^{2}}\) intersect at point (1, 1).

Step 1: Plotting the Graphs

The first step is to visualize the graphs of the two functions \(y=\frac{1}{x}\) and \(y=\frac{1}{x^{2}}\). This can be done using any graph plotting tool.

Step 2: Finding the Points of Intersection

The points of intersection of the two graphs can be found by equating the two equations \(y=\frac{1}{x}\) and \(y=\frac{1}{x^{2}}\). Doing so gives, \(\frac{1}{x}=\frac{1}{x^{2}}\), which simplifies to \(x^{2}=1\). Solving this equation gives \(x=1\) and \(x=-1\).

Step 3: Verifying the Points of Intersection

To ensure that the points \(x=1\) and \(x=-1\) are truly the points of intersection, substitute them back into the original equations. For \(x=1\), \(y=\frac{1}{1}=1\) and \(y=\frac{1}{(1)^{2}}=1\) for both equations. For \(x=-1\), \(y=\frac{1}{-1}=-1\) and \(y=\frac{1}{(-1)^{2}}=1\) for the first and second equation respectively, indicating different y values. So, the graphs intersect only at \(x=1\), \(y=1\) and not at \(x=-1\).

Key Concepts

Inverse Functions

Understanding inverse functions is crucial in comparing graphs and finding their intersections. An inverse function effectively reverses the roles of inputs and outputs of the original function. For a function \(f(x)\), its inverse is denoted \(f^{-1}(x)\). Inverse functions undo each other, which means if you apply a function and its inverse in succession, you end up with the original quantity. Graphically, if one were to plot a function and its inverse on the same set of axes, they should reflect each other across the line \(y = x\). This knowledge is particularly useful when we find intersections, as it helps in analyzing how one function's transformation mirrors the other, forming points of intersections. Explore inverse functions through examples and by sketching their reflections to grasp their properties better.

Rational Functions

Rational functions are defined by the ratio of two polynomials. In simpler terms, if a function can be represented as \(\frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials, the function is rational.In our exercise, both functions \(y=\frac{1}{x}\) and \(y=\frac{1}{x^{2}}\) are rational. These functions often have

• vertical asymptotes, which are lines that the graph approaches but never crosses, typically where the denominator is zero;
• horizontal asymptotes, which show the end behavior of a graph as \(x\) goes to positive or negative infinity.

When solving the given exercise, it was important to understand these characteristics to find intersections correctly, focusing on valid \(x\) values that don’t result in undefined expressions.

Solving Equations

Solving equations is foundational in finding intersection points of graphs. In our problem, we equated two rational functions: \(\frac{1}{x} = \frac{1}{x^2}\). By multiplying both sides by \(x^2\) to eliminate the denominators, we streamlined our equation to \(x^2 = 1\). Solving this results in solutions \(x = 1\) and \(x = -1\). It’s crucial to substitute these back into the original equations to verify whether they are actual intersection points.It helps to remember these steps when solving similar equations:

• Bring terms to a common denominator when possible.
• Simplify systematically to make the solution easier.
• Always verify proposed solutions in the context of the original equations.

These techniques ensure your solutions are valid and accurate, contributing to a better understanding of graph intersections.