Select the correct answer.
Consider functions [tex]$f$[/tex] and [tex]$g$[/tex].
[tex]$\begin{array}{l}
f(x)=(x+1)^2-2 \\
g(x)=\frac{2}{x}
\end{array}$[/tex]
Using graphing, what is the approximate solution, or solutions, to the equation [tex]$f(x)=g(x)$[/tex]?
A. [tex]$x=-1$[/tex]
B. [tex]$x \approx 2.7$[/tex]
C. [tex]$x \approx-2.2$[/tex] and [tex]$x \approx-0.6$[/tex] and [tex]$x \approx 0.8$[/tex]
D. [tex]$x=-2$[/tex] and [tex]$x=-1$[/tex] and [tex]$x=1$[/tex]
Answer (1)
Set the two functions equal to each other: ( x + 1 ) 2 − 2 = x 2 .
Expand and simplify the equation to get a cubic equation: x 3 + 2 x 2 − x − 2 = 0 .
Find the roots of the cubic equation, which are the solutions to the original equation: x = − 2 , − 1 , 1 .
The approximate solutions to the equation f ( x ) = g ( x ) are x = − 2 , x = − 1 , and x = 1 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = ( x + 1 ) 2 − 2 and g ( x ) = x 2 , and we want to find the approximate solutions to the equation f ( x ) = g ( x ) using graphing. This means we need to find the x-values where the graphs of the two functions intersect.
Setting up the Equation To find the solutions, we set the two functions equal to each other: ( x + 1 ) 2 − 2 = x 2
Expanding the Equation We can solve this equation algebraically. First, expand the left side: x 2 + 2 x + 1 − 2 = x 2 x 2 + 2 x − 1 = x 2
Clearing the Fraction Multiply both sides by x to get rid of the fraction: x ( x 2 + 2 x − 1 ) = 2 x 3 + 2 x 2 − x = 2 x 3 + 2 x 2 − x − 2 = 0
Finding the Roots Now we need to find the roots of the cubic equation x 3 + 2 x 2 − x − 2 = 0 . We can use the python_calculation_tool to find the roots of this equation. The approximate solutions are x = − 2 , x = − 1 , and x = 1 .
Comparing with Options Now, let's check these solutions with the given options. The solutions we found are x = − 2 , x = − 1 , and x = 1 . Comparing this with the given options, we see that the correct answer is: x = − 2 and x = − 1 and x = 1
Examples
Consider the scenario where you are comparing the performance of two different investment strategies. Function f ( x ) represents the return from one strategy, and function g ( x ) represents the return from another strategy. By finding the solutions to the equation f ( x ) = g ( x ) , you can determine the points in time (x-values) when the two strategies yield the same return. This helps in making informed decisions about when to switch between strategies or combine them for optimal results. Understanding the intersection points of these functions is crucial for effective investment management.
