Functions, f and g are given by f(x)= 3+ cos x and g(x) = 2x, x is a real number. Determine the value of c for which f(g(x))= g(f(x)) where 0\leq x<2\pi

Question

Functions, f and g are given by f(x)= 3+ cos x and g(x) = 2x, x is a real number. Determine the value of c for which f(g(x))= g(f(x)) where 0
`\leq` x<2
`\pi`

Answer 1

9514 1404 393

Answer:

x = π

Explanation:

You want f(g(x)) = g(f(x)):

3 +cos(2x) = 2(3 +cos(x))

cos(2x) -2cos(x) = 3 . . . . . . . rearrange

2cos(x)²-1 -2cos(x) = 3 . . . . . use an identity for cos(2x)

2(c² -c -2) = 0 . . . . . . . . . . . . substitute c = cos(x)

(c -2)(c +1) = 0 . . . . . . . . . . . factor

c = 2 (not possible)

c = -1 = cos(x) . . . . . true for x = π

The value of x that makes f(g(x)) = g(f(x)) is x = π.

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Additional comment

The substitution c=cos(x) just makes the equation easier to write and the form of it easier to see. There is really no other reason for making any sort of substitution. In the end, the equation is quadratic in cos(x), so is solved by any of the usual methods of solving quadratics.