This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's an explanation of the provided solution for finding the composite function (f f)(x) and its domain. Given the function f(x) = (4x)/(x-1). Step 1: Find the composite function (f f)(x). The composite function (f f)(x) means applying the function f to f(x). So, (f f)(x) = f(f(x)). To calculate this, we substitute the entire expression for f(x) into f(x) wherever x appears. (f f)(x) = f((4x)/(x-1)) Substitute (4x)/(x-1) for x in the definition of f(x): (f f)(x) = (4(4x)/(x-1))((4x)/(x-1))-1 Step 2: Simplify the expression. First, multiply the numerator: (4(4x)/(x-1))((4x)/(x-1))-1 = (16x)/(x-1)(4x)/(x-1)-1 Next, simplify the denominator by finding a common denominator: (4x)/(x-1)-1 = (4x)/(x-1) - (x-1)/(x-1) = (4x - (x-1))/(x-1) = (4x - x + 1)/(x-1) = (3x+1)/(x-1) Now, substitute this back into the complex fraction: (f f)(x) = (16x)/(x-1)(3x+1)/(x-1) To simplify a complex fraction, multiply the numerator by the reciprocal of the denominator: (f f)(x) = (16x)/(x-1) · (x-1)/(3x+1) Cancel out the common term (x-1): (f f)(x) = (16x)/(3x+1) Step 3: Determine the domain of (f f)(x). For a composite function (f f)(x) to be defined, two conditions must be met: 1. The input x must be in the domain of the inner function, f(x). The function f(x) = (4x)/(x-1) is undefined when its denominator is zero, so x-1 ≠ 0, which means x ≠ 1. 2. The output of the inner function, f(x), must be in the domain of the outer function, f. This means f(x) cannot take the value that makes f(x) undefined, which is 1. So, f(x) ≠ 1. Substitute the expression for f(x): (4x)/(x-1) ≠ 1 Multiply both sides by (x-1) (assuming x ≠ 1): 4x ≠ 1(x-1) 4x ≠ x-1 Subtract x from both sides: 3x ≠ -1 Divide by 3: x ≠ -(1)/(3) Additionally, the denominator of the simplified composite function (f f)(x) = (16x)/(3x+1) cannot be zero. 3x+1 ≠ 0 3x ≠ -1 x ≠ -(1)/(3) This condition is already covered by the second point above. Combining these conditions, the domain of (f f)(x) is all real numbers x such that x ≠ 1 and x ≠ -(1)/(3). What's next?