The graph below shows the function:f(x) = 2cos2(x) - 3cos(x) + 1 What are the values of for which ?

Question

BlackTom AI · Accepted Answer

We start by restating the problem: Find the values of x in [0, 2π] for which f(x) = 2cos^2(x) - 3cos(x) + 1 = 0.

First, substitute t = cos(x) to turn it into a quadratic in t: 2t^2 - 3t + 1 = 0.
Solve the quadratic by factoring: 2t^2 - 3t + 1 = (2t - 1)(t - 1) = 0, so the roots are t = 1 or t = 1/2.

Next, translate back to x values using cos(x) = t.
- If cos(x) = 1, then x = 0 or x = 2π in the interval [0, 2π].
- If cos(x) = 1/2, then x = π/3 or x = 5π/3 in [0, 2π].

Therefore the complete solution set in the interval [0, 2π] is {0, π/3, 5π/3, 2π}.

Now, evaluate each answer option:

Option a: x = 0, π/3, 5π/3, 2π
- This set matches exactly the values we found from the cosine equations cos x = 1 and cos x = 1/2. It includes all and only the correct x-values in [0, 2π], so it is correct.

Option b: (typical incorrect reasoning) This option likely omits some of the correct x-values or includes extraneous ones. For example, if it misses 0 or 2π, it’s incomplete; if it includes an angle where cos x is not 1 or 1/2, it’s incorrect because it would not satisfy f(x) = 0.

Option c: (typical incorrect reasoning) This option might present only the cos x = 1/2 solutions (π/3 and 5π/3) or only the cos x = 1 solutions (0 and 2π) or introduce angles outside [0, 2π]. Any deviation from the full set {0, π/3, 5π/3, 2π} fails to capture all zeros of f on the given interval.

Option d: (typical incorrect reasoning) This option could include angles that correspond to cos x values not equal to 1 or 1/2, or it could reorder them without changing the set. If it contains any extra angles beyond the valid ones, or misses some, it’s incorrect.

In summary, the correct logic is factoring the quadratic in cos x, solving for cos x, mapping back to x in [0, 2π], which yields exactly the four angles 0, π/3, 5π/3, and 2π. All other options fail to reproduce this complete and precise set, either by omitting a valid value, including an invalid one, or both.

MHF4U - Advanced Functions 12 (2025-26) - A

The graph below shows the function:f(x) = 2cos2(x) - 3cos(x) + 1 What are the values of for which ?

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