Consider the functions \( f(x) = x^2 – 7x + 10 \) and \( g(x) = \ln(x) \). We want to find the points of intersection of these two functions, which requires solving the equation \( x^2 – 7x + 10 = \ln(x) \). Remember that the domain of the logarithmic function \( g(x) = \ln(x) \) is \( x > 0 \).
(a) Write down the equation that needs to be solved to find the x-coordinates of the intersection points of \( f(x) \) and \( g(x) \). State the domain restriction that must be considered.
(b) Find the approximate solutions to the equation \( x^2 – 7x + 10 = \ln(x) \). Give your answers rounded to two decimal places.
(c) (For HL students) Verify that the approximate solutions found in part (b) are valid solutions to the equation by substituting them back into \( f(x) \) and \( g(x) \). Comment on the accuracy of your approximations and ensure that the domain restriction is satisfied.
Question
Write down the equation to find the intersection points of \( f(x) = x^2 – 7x + 10 \) and \( g(x) = \ln(x) \), and state the necessary domain restriction.
Hint
Hint
Hint
Set \(f(x) = g(x)\) and consider the domain of \(\ln(x)\).
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Complete Answer
Equation: \( x^2 – 7x + 10 = \ln(x) \)
Domain restriction: \( x > 0 \)
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Question
Solve the equation \( x^2 – 7x + 10 = \ln(x) \) approximately. Round your answers to two decimal places.
Hint
Hint
Hint
Use a graphical calculator to graph \( y = x^2 – 7x + 10 \) and \( y = \ln(x) \) and find the x-coordinates of their intersection points.
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Complete Answer
The approximate solutions are \( x \approx 1.82 \) and \( x \approx 5.49 \).
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For HL students: Verify that the solutions \( x \approx 1.82 \) and \( x \approx 5.49 \) are valid approximate solutions by substituting them into \( f(x) \) and \( g(x) \). Comment on the accuracy and domain considerations.
Hint
Hint
Hint
Calculate \(f(1.82)\), \(g(1.82)\), \(f(5.49)\), and \(g(5.49)\) and compare the values. Check if both solutions are in the domain \(x > 0\).
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For \( x \approx 1.82 \): \( f(1.82) \approx 0.57 \), \( g(1.82) \approx 0.60 \).
For \( x \approx 5.49 \): \( f(5.49) \approx 1.71 \), \( g(5.49) \approx 1.70 \).
Both solutions are valid as they are in the domain \( x > 0 \) and give close values for \( f(x) \) and \( g(x) \).
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