The function f:R-->R is given by f(x)=4x-3. Solve the equation f[f(x)]-f(1)=0

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x) = 4x - 3. We have to solve the equation f[f(x)] - f(1) = 0

f[f(x)] - f(1) = 0

=> f(4x - 3) - 4*1 + 3 = 0

=> 4(4x - 3) - 3 - 4 + 3 = 0

=> 16x - 12 - 4 = 0

=> 16x - 16 = 0

=> 16x = 16

=> x = 1

The solution of the given equation is x = 1.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To solve the equation, we need to determine the term f(f(x)), that is the result of composition of functions, such as: (fof)(x) = f(f(x)).

f(f(x)) = 4*f(x) - 3

f(f(x)) = 4*(4x-3) - 3

f(f(x)) = 16x - 12 - 3

f(f(x)) = 16x - 15 (1)

Now, we'll determine the value of the function at x = 1:

f(1) = 4*1 - 3

f(1) = 1 (2)

We'll replace (1) and (2) in the equation that has to be solved:

16x - 15 - 1 = 0

We'll combine like terms:

16x - 16 = 0

We'll divide by 16 both sides:

x - 1 = 0

x = 1

The solution of the equation f(f(x)) - f(1) = 0 is x = 1.

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