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
a) Step 1: Find the inverse function f^-1(x). Given f(x) = (x)/(x-3). Let y = f(x), so y = (x)/(x-3). To find the inverse, swap x and y: x = (y)/(y-3) Now, solve for y: x(y-3) = y xy - 3x = y xy - y = 3x y(x-1) = 3x y = (3x)/(x-1) So, f^-1(x) = (3x)/(x-1). Step 2: State the domain of f^-1(x). The domain of f^-1(x) is restricted when the denominator is zero. x-1 ≠ 0 x ≠ 1 The domain of f^-1(x) is \x R x ≠ 1\. The inverse function is f^-1(x) = (3x)/(x-1) with domain \x R x ≠ 1\. b) Step 1: Determine the composite function fg(x). Given f(x) = (x)/(x-3) and g(x) = sqrt(x+2). The composite function fg(x) is f(g(x)): fg(x) = f(sqrt(x+2)) = sqrt(x+2)sqrt(x+2)-3 Step 2: Find the domain of fg(x). For g(x) = sqrt(x+2) to be defined, x+2 0 x -2. For f(g(x)) to be defined, the denominator sqrt(x+2)-3 cannot be zero. sqrt(x+2)-3 ≠ 0 sqrt(x+2) ≠ 3 x+2 ≠ 3^2 x+2 ≠ 9 x ≠ 7 Combining these conditions, the domain of fg(x) is \x R x -2 and x ≠ 7\. Step 3: Find the range of fg(x). Let u = g(x) = sqrt(x+2). From the domain of fg(x), we know x -2 and x ≠ 7. This implies u = sqrt(x+2) sqrt(-2+2) = 0. Also, u = sqrt(x+2) ≠ sqrt(7+2) = sqrt(9) = 3. So, the range of u is [0, ) \3\. Now we need to find the range of f(u) = (u)/(u-3) for u [0, ) \3\. Let y = (u)/(u-3). To find the range, we can express u in terms of y: y(u-3) = u yu - 3y = u yu - u = 3y u(y-1) = 3y