Question

Use a graphing calculator to determine where \(f(x)=g(x)\). $$f(x)=\frac{2}{x}+1, g(x)=x^2+1$$

Short answer

The two functions \(f(x)\) and \(g(x)\) intersect at \(x = \sqrt[3]{2}\).

Step-by-step solution

Set the Functions Equal

First, set the two functions equal to each other to find where they intersect. This results in the equation \(\frac{2}{x} + 1 = x^2 + 1\).

Simplify the Equation

Next, subtract 1 from both sides of the equation to isolate terms. The equation becomes \(\frac{2}{x} = x^2\).

Multiply by x

To remove the denominator, multiply both sides of the equation by \(x\). The equation becomes \(2 = x^3\).

Solve the Equation

Now, to find the exact value of \(x\), take the cubic root of both sides. Simplifying results in \(x = \sqrt[3]{2}\).

Verify the Solution

Use a graphing calculator, plot both functions \(f(x)\) and \(g(x)\), and verify that they intersect at \(x = \sqrt[3]{2}\). Alternatively, substitute the value of \(x\) back into the original equations to verify the solution.

Key concepts

Graphing Calculator

A graphing calculator is an indispensable tool for visualizing functions and their interactions. It allows students to plot mathematical expressions, such as equations or inequalities, and observe where they intersect. When you're tasked with finding the intersection of two functions, like in our example where we have functions f(x) and g(x), a graphing calculator provides a visual confirmation of the solution you've algebraically determined.

To use a graphing calculator for this purpose, you would enter the function formulas separately and then use the device's functionalities to display their graphs on the same set of axes. You'd look for points where the graphs cross, which represent the values of x where f(x) = g(x). This graphical viewpoint can be particularly helpful if the functions are complex, or if their intersection isn't obvious through algebraic manipulation alone.

Function Intersection

The concept of a function intersection refers to the point(s) where two different functions have the same value for both x (the input) and y (the output). It's essentially where the graphs of the functions meet on a coordinate plane. In the given problem, we're interested in finding the intersection of the functions represented by f(x) and g(x).

Algebraically, to find function intersections, you'll often set the functions equal to each other and solve for x, as demonstrated in our step-by-step solution. Once we have the x value of the intersection, we can input it back into either of the original functions to find the corresponding y value, confirming the intersection point.

Cubic Root

In mathematics, the cubic root of a number is a value that, when multiplied by itself three times, gives the original number. It's denoted as \( \sqrt[3]{x} \). In the context of our exercise, we're dealing with finding the cubic root of 2 to solve the equation \(2 = x^3\).

Taking the cubic root of both sides of this equation gives us \(x = \sqrt[3]{2}\), which is the exact value of x where our two functions intersect. Understanding how to calculate cubic roots—either by hand, with a calculator, or by using properties of exponents—is essential for solving such algebraic equations.

Algebraic Manipulation

The process of algebraic manipulation involves rearranging and simplifying equations to solve for unknown variables. It's a critical skill in algebra that enables us to solve complex problems systematically. In the example provided, we manipulated the equation step by step, first setting the functions equal, then isolating terms, and eventually removing fractions and solving a cubic equation.

Each step requires careful consideration to ensure that the actions are mathematically valid and that they bring us closer to the solution. Mastery of algebraic manipulation is achieved through practice and understanding of algebraic principles, which can then be applied to a broad range of problems beyond just finding the intersection of functions.