Question

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

Short answer

The solutions can be approximated visually on the graphing calculator where the two functions intersect. Since this is a cubic equation, there could be up to 3 solutions depending on the location of the points of intersection. For a precise numerical solution, one could use numerical methods or a calculator's intersection-finding feature if available.

Step-by-step solution

Expressing f(x) in terms of g(x)

Take the two functions and set them as equal to each other. \[\frac{1}{x}+1 = x^2\]

Algebraic Rearrangement

Rearrange the equation to express it in the x terms. This results in a cubic equation. \[ x^3 - x +1 = 0\]

Graphing and Identifying Solutions

Use a graphing calculator to graph both equations \( y = \frac{1}{x}+1,\) and \( y= x^2\). Then identify where the two graphs intersect, which is the solution to the equation.

Key concepts

Algebraic Equations

An algebraic equation is essentially a mathematical statement that asserts the equality of two expressions. These expressions can be made up in a variety of ways, including numbers, variables (like x), and operations such as addition (+), subtraction (-), multiplication (*), and division (/).

When tackling algebraic equations, our goal is often to find the value of the unknown variable that makes the equation true. For the given exercise, the equation is set up by equating two functions, f(x) and g(x), leading to \[\frac{1}{x} + 1 = x^2\]. This initial step sets the stage for solving the equation. Since we’re dealing with functions, which are essentially equations that assign an output to every input, finding the intersection points means finding the inputs (x values) at which their outputs are equal — a foundational concept in solving algebraic equations.

Cubic Equations

Cubic equations are a specific type of algebraic equation characterized by at least one term raised to the third power. Generally, cubic equations take the form \[ax^3+bx^2+cx+d=0\], where a, b, c, and d are constants, and a is nonzero.

This kind of equation introduces complexities not found in linear or quadratic equations, including multiple real roots, or a combination of real and complex roots. In the exercise provided, algebraic rearrangement transforms the given equation into a cubic equation \[x^3 - x + 1 = 0\]. Solving cubic equations often involves advanced techniques such as synthetic division, factoring, or utilizing the Rational Root Theorem. However, when the solutions are not straightforward to find algebraically, graphical methods become a handy tool, effectively turning a daunting algebraic challenge into a visual problem-solving exercise.

Graphing Techniques

Graphing techniques are instrumental in visualizing the relationship between variables in mathematical functions and equations. They allow us to convey complex algebraic concepts through a graphical representation, adding an intuitive dimension to understanding and solving equations.

Using a graphing calculator, one can input different algebraic equations to produce their corresponding graphs. This visualization helps in identifying key features such as intercepts, turning points, and areas of increasing or decreasing values. When plotting the functions \[y = \frac{1}{x}+1\], and \[y= x^2\], the graphing calculator provides a visual intersection of the two graphs where the value of y from both equations is equal. This point of intersection represents the solution set of the initial problem, as it points to the x-values that satisfy both f(x) and g(x). Understanding these graphical representations is not only crucial for finding the points of intersection but also enhances comprehending the behavior of different algebraic equations.