$\begin{array}{l} f(x)=x^3-3 x-3 \\ g(x)=\sqrt{x+1} \end{array}$ What is the approximate solution to the equation $f(x)=g(x)$? Use the graph as a starting point $x \approx \frac{37}{16}$ $x \approx \frac{15}{8}$ $x \approx \frac{21}{8}$ $x \approx \frac{77}{16}$
Answer (2)
Evaluate f ( x ) and g ( x ) for each given x value.
Calculate the absolute difference ∣ f ( x ) − g ( x ) ∣ for each x .
Compare the absolute differences to find the minimum.
The x value with the smallest difference is the approximate solution: 16 37 .
Explanation
Problem Analysis We are given two functions, f ( x ) = x 3 − 3 x − 3 and g ( x ) = x + 1 , and we want to find the approximate solution to the equation f ( x ) = g ( x ) among the given options: x ≈ 16 37 , x ≈ 8 15 , x ≈ 8 21 , x ≈ 16 77 . To do this, we will evaluate f ( x ) and g ( x ) for each of the given values of x and find the value of x for which f ( x ) is closest to g ( x ) .
Evaluating Functions Let's evaluate f ( x ) and g ( x ) for each given value of x .
Calculations for x = 37/16 For x = 16 37 = 2.3125 :
f ( 16 37 ) = ( 16 37 ) 3 − 3 ( 16 37 ) − 3 = 2.312 5 3 − 3 ( 2.3125 ) − 3 = 12.3439 − 6.9375 − 3 = 2.4064 g ( 16 37 ) = 16 37 + 1 = 16 37 + 16 16 = 16 53 = 3.3125 ≈ 1.8200 ∣ f ( 16 37 ) − g ( 16 37 ) ∣ = ∣2.4290 − 1.8200∣ = 0.6090
Calculations for x = 15/8 For x = 8 15 = 1.875 :
f ( 8 15 ) = ( 8 15 ) 3 − 3 ( 8 15 ) − 3 = 1.87 5 3 − 3 ( 1.875 ) − 3 = 6.5918 − 5.625 − 3 = − 2.0332 g ( 8 15 ) = 8 15 + 1 = 8 15 + 8 8 = 8 23 = 2.875 ≈ 1.6956 ∣ f ( 8 15 ) − g ( 8 15 ) ∣ = ∣ − 2.0332 − 1.6956∣ = 3.7288
Calculations for x = 21/8 For x = 8 21 = 2.625 :
f ( 8 21 ) = ( 8 21 ) 3 − 3 ( 8 21 ) − 3 = 2.62 5 3 − 3 ( 2.625 ) − 3 = 18.1191 − 7.875 − 3 = 7.2441 g ( 8 21 ) = 8 21 + 1 = 8 21 + 8 8 = 8 29 = 3.625 ≈ 1.9039 ∣ f ( 8 21 ) − g ( 8 21 ) ∣ = ∣7.2129 − 1.9039∣ = 5.3090
Calculations for x = 77/16 For x = 16 77 = 4.8125 :
f ( 16 77 ) = ( 16 77 ) 3 − 3 ( 16 77 ) − 3 = 4.812 5 3 − 3 ( 4.8125 ) − 3 = 111.342 − 14.4375 − 3 = 93.9045 g ( 16 77 ) = 16 77 + 1 = 16 77 + 16 16 = 16 93 = 5.8125 ≈ 2.4110 ∣ f ( 16 77 ) − g ( 16 77 ) ∣ = ∣94.0208 − 2.4110∣ = 91.6098
Finding the Closest Solution Comparing the absolute differences between f ( x ) and g ( x ) for each value of x , we have:
For x = 16 37 , ∣ f ( x ) − g ( x ) ∣ = 0.6090
For x = 8 15 , ∣ f ( x ) − g ( x ) ∣ = 3.7288
For x = 8 21 , ∣ f ( x ) − g ( x ) ∣ = 5.3090
For x = 16 77 , ∣ f ( x ) − g ( x ) ∣ = 91.6098
The smallest absolute difference is 0.6090 , which occurs when x = 16 37 . Therefore, the approximate solution to the equation f ( x ) = g ( x ) is x ≈ 16 37 .
Final Answer The approximate solution to the equation f ( x ) = g ( x ) is x ≈ 16 37 .
Examples
In engineering, finding the intersection of two functions is crucial for determining system stability. For instance, if f ( x ) represents the power output of a device and g ( x ) represents the power consumption, the point where f ( x ) = g ( x ) indicates the equilibrium state. Approximating this point, as we did here, helps engineers ensure the device operates efficiently without overloading. This method is also used in economics to find market equilibrium, where supply and demand functions intersect, determining the optimal price and quantity.
The approximate solution to the equation f ( x ) = g ( x ) is found by evaluating both functions at the candidate values. The smallest absolute difference occurs at x = 16 37 , so this is the approximate solution. Thus, the answer is x ≈ 16 37 .
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