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f(x) is a non constant linear function such that f(f(f(f(1)))) is equal to f(f(1)). What is the value of f(3)+2f(0) if it is given that the angle between the function and the X axis is not equal to 450
 
Enter -1 if the answer can't be determined.
    Correct answer is '3'. Can you explain this answer?
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    f(x) is a non constant linear function such that f(f(f(f(1)))) is equa...
    Let the function be f(x) = mx+c, as it is given that the function, is linear, here 'm' is the slope of the function
    f(f(x)) = m2x+c+mc
    similarly f(f(f(x))) = f(m2x + c + m3x+c+mc+m2c
    and f(f(f(f(x)))) = m4x + c + mc + m2c + m3c
    f(f(f(f(1)))) = f(f(1)), Thus, m4+c+mc+m2c+m3c = m2+c+mc
    On rearranging we get m4+m2c+m3c = m2
    Since the function is not constant we can say that m ≠ 0
    Dividing both side by m2 and shifting all term to one side we get
    m2 + c + mc − 1 = 0
    (m2−1)+(c+mc) = 0
    (m+1)(m−1)+c(1+m)=0
    (m+1)(m+c−1) = 0
    Since the function does not make 450 with x-axis the value of m can not be 1 or -1
    Thus m+c-1 = 0 or m+c = 1....(I)
    f(3) = 3m+c and f(0) = c
    f(3)+2f(0) = 3m+c+2c 
    = 3m+c+2c = 3(m+c) = 3 (inputting m+c = 1 from (I))
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    f(x) is a non constant linear function such that f(f(f(f(1)))) is equa...
    Given information:
    - f(x) is a non-constant linear function.
    - f(f(f(f(1)))) = f(f(1)).
    - The angle between the function and the x-axis is not equal to 450.

    To find:
    The value of f(3) + 2f(0).

    Solution:
    Let's assume the general form of a linear function as f(x) = mx + c, where m is the slope and c is the y-intercept.

    Step 1: Find f(1):
    Since f(x) is a linear function, we can substitute x = 1 in f(x) = mx + c to find f(1).
    f(1) = m(1) + c = m + c

    Step 2: Find f(f(1)):
    Substituting f(1) in f(x), we get:
    f(f(1)) = mf(1) + c

    Step 3: Find f(f(f(1))):
    Substituting f(f(1)) in f(x), we get:
    f(f(f(1))) = mf(f(1)) + c = m(mf(1) + c) + c = m^2f(1) + mc + c

    Step 4: Find f(f(f(f(1)))):
    Substituting f(f(f(1))) in f(x), we get:
    f(f(f(f(1)))) = mf(f(f(1))) + c = m(m^2f(1) + mc + c) + c = m^3f(1) + m^2c + mc + c

    Given that f(f(f(f(1)))) = f(f(1)), we can equate the expressions:
    m^3f(1) + m^2c + mc + c = mf(1) + c

    Step 5: Simplify the equation:
    Rearranging the terms, we get:
    m^3f(1) + m^2c + mc + c - mf(1) - c = 0

    Simplifying further, we get:
    m^3f(1) + m^2c + mc - mf(1) = 0

    Step 6: Determine the coefficients:
    Since f(x) is a non-constant linear function, the coefficient of f(1) must be non-zero. Therefore, m^3 - m = 0.

    This equation can be factored as m(m^2 - 1) = 0.

    The solutions for m are m = 0 and m = ±1.

    Step 7: Analyzing the solutions:
    If m = 0, the function becomes f(x) = c, which is a constant function. However, it is given that f(x) is a non-constant linear function. Therefore, m = 0 is not a valid solution.

    If m = ±1, the function can be written as f(x) = ±x + c.

    Step 8: Analyzing the angle between the function and the x-axis:
    Since the angle between the function and the x-axis is not equal to 450, the slope (m) of the
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    f(x) is a non constant linear function such that f(f(f(f(1)))) is equal to f(f(1)). What is the value of f(3)+2f(0) if it is given that the angle between the function and the X axis is not equal to 450Enter -1 if the answer cant be determined.Correct answer is '3'. Can you explain this answer?
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