Select the correct answer.
Consider functions [tex]$f$[/tex] and [tex]$g$[/tex].
[tex]
\begin{array}{l}
f(x)=\frac{1}{4} x^4-2 x^2+4 \\
g(x)=2 \sqrt{x-2}
\end{array}
[/tex]
Using graphing, what is the approximate solution, or solutions, of [tex]$f(x)=g(x)$[/tex]?
A. [tex]$x=2$[/tex] and [tex]$x \approx 2.5$[/tex]
B. [tex]$x=0$[/tex] and [tex]$x \approx 1.5$[/tex]
C. [tex]$x=0$[/tex] and [tex]$x \approx 0.6$[/tex]
D. [tex]$x \approx 2.7$[/tex]
Answer (2)
We are given two functions f ( x ) and g ( x ) and need to find the solutions to f ( x ) = g ( x ) using graphing.
Graph the two functions and find their intersection points.
The x-coordinates of the intersection points are the solutions to the equation.
The approximate solutions are x = 2 and x ≈ 2.5 , so the answer is x = 2 and x ≈ 2.5 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 4 1 x 4 − 2 x 2 + 4 and g ( x ) = 2 x − 2 , and we want to find the approximate solution(s) of 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 intersection points, we need to solve the equation f ( x ) = g ( x ) , which is 4 1 x 4 − 2 x 2 + 4 = 2 x − 2 . This equation is difficult to solve analytically, so we will use the graphing approach as suggested.
Graphing the Functions Using a graphing tool or a calculator, we can graph the two functions f ( x ) and g ( x ) . By observing the graphs, we can identify the points where the two curves intersect. The x-coordinates of these intersection points are the solutions to the equation f ( x ) = g ( x ) .
Finding the Intersection Points By using a root-finding tool, we find that the approximate solutions are x = 2 and x ≈ 2.533 . Comparing these values with the given options, we see that the closest option is x = 2 and x ≈ 2.5 .
Examples
Imagine you are designing a rollercoaster. The function f ( x ) represents the height of one part of the track, and g ( x ) represents the height of another part. To ensure a smooth transition between the two parts, you need to find the points where the heights are equal, i.e., where f ( x ) = g ( x ) . Finding these intersection points helps you design a safe and enjoyable rollercoaster ride.
By graphing the functions f ( x ) and g ( x ) , we can see their intersection points, which are approximately at x = 2 and x ≈ 2.5 . Therefore, the correct answer is option A: x = 2 and x ≈ 2.5 .
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