Select the correct answer.
Consider functions [tex]$f$[/tex] and [tex]$g$[/tex].
[tex]
\begin{array}{l}
f(x)=x^4-16 x^3+94 x^2-240 x+225 \\
g(x)=\frac{1}{x-4}-1
\end{array}
[/tex]
Using graphing, what is the solution, or solutions, of [tex]$f(x)=g(x)$[/tex]?
A. [tex]$x=0 ; x=5$[/tex]
B. [tex]$x=3$[/tex]
C. [tex]$x=0 ; x=3$[/tex]
D. [tex]$x=5$[/tex]
Answer (1)
Set up the equation f ( x ) = g ( x ) .
Find the roots of the equation f ( x ) − g ( x ) = 0 .
Verify the root by plugging it back into the original equation.
The solution to f ( x ) = g ( x ) is x = 5 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 4 − 16 x 3 + 94 x 2 − 240 x + 225 and g ( x ) = x − 4 1 − 1 , and we want to find the solution(s) 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 need to solve the equation f ( x ) = g ( x ) , which is equivalent to finding the roots of the equation f ( x ) − g ( x ) = 0 . So, we are looking for the values of x such that x 4 − 16 x 3 + 94 x 2 − 240 x + 225 − ( x − 4 1 − 1 ) = 0 .
Finding the Root Using a numerical method (or a graphing calculator), we find that there is a root near x = 5 . We can verify this by plugging in x = 5 into the original equation:
f ( 5 ) = ( 5 ) 4 − 16 ( 5 ) 3 + 94 ( 5 ) 2 − 240 ( 5 ) + 225 = 625 − 2000 + 2350 − 1200 + 225 = 0
g ( 5 ) = 5 − 4 1 − 1 = 1 1 − 1 = 1 − 1 = 0
Since f ( 5 ) = g ( 5 ) = 0 , x = 5 is indeed a solution.
Final Answer The approximate real-valued roots calculation tool found a root at approximately x = 5 . Therefore, the solution to f ( x ) = g ( x ) is x = 5 .
Examples
In engineering, finding the intersection points of two functions is crucial for determining system stability. For example, f ( x ) could represent the output of a control system, and g ( x ) could represent a desired setpoint. The solution to f ( x ) = g ( x ) would then indicate when the system's output matches the desired setpoint, ensuring proper control and stability. This is also applicable in economics when determining market equilibrium where supply and demand curves intersect.
