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.
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Step 1: State the domain of each function. For f(x) = 1 - 2x: This is a linear function, which is defined for all real numbers. The domain of f is R or (-, ). For g(x) = (1)/(x+3): This is a rational function, which is defined when its denominator is not zero. So, x+3 ≠ 0 x ≠ -3. The domain of g is \x R x ≠ -3\ or (-, -3) (-3, ). Step 2: Show that g(a) = g(b) a = b and state the conclusion. Assume g(a) = g(b). (1)/(a+3) = (1)/(b+3) Cross-multiply: b+3 = a+3 Subtract 3 from both sides: b = a Since g(a) = g(b) implies a = b, the function g(x) is injective (or one-to-one). Step 3: Find the inverse function g^-1(x). Let y = g(x) = (1)/(x+3). To find the inverse, swap x and y: x = (1)/(y+3) Multiply both sides by (y+3): x(y+3) = 1 xy + 3x = 1 Isolate y: xy = 1 - 3x y = (1 - 3x)/(x) So, the inverse function is g^-1(x) = (1 - 3x)/(x). Step 4: Find the composite function f(g(x)). Given f(x) = 1 - 2x and g(x) = (1)/(x+3). Substitute g(x) into f(x): f(g(x)) = 1 - 2((1)/(x+3)) f(g(x)) = 1 - (2)/(x+3) Combine terms with a common denominator: f(g(x)) = (x+3)/(x+3) - (2)/(x+3) f(g(x)) = (x+3-2)/(x+3) f(g(x)) = (x+1)/(x+3) Step 5: Solve the equation f(g(x)) = g^-1(x). Set the expressions for f(g(x)) and g^-1(x) equal: (x+1)/(x+3) = (1-3x)/(x) Cross-multiply: x(x+1) = (1-3x)(x+3) Expand both sides: x^2 + x = x + 3 - 3x^2 - 9x x^2 + x = -3x^2 - 8x + 3 Rearrange into a standard quadratic equation ax^2 + bx + c = 0: x^2 + 3x^2 + x + 8x - 3 = 0 4x^2 + 9x - 3 = 0 Step 6: Use the quadratic formula to find the values of x. For 4x^2 + 9x - 3 = 0, we have a=4, b=9, c=-3. The quadratic formula is x = -b ± sqrt(b^2 - 4ac)2a. x = -9 ± sqrt(9^2 - 4(4)(-3))2(4) x = -9 ± sqrt(81 + 48)8 x = -9 ± sqrt(129)8 The answer is left in surd form. i. State the domain of each function The domain of f is R. The domain of g is \x R x ≠ -3\. ii. Show that g(a) = g(b) a = b, what conclusion can you draw from the result As shown in Step 2, g(a) = g(b) a = b. Conclusion: The function g(x) is injective (or one-to-one). iii. Find the inverse function g^-1 of g and hence the value of x for which f(g(x)) = g^-1x leaving your answer in surd form. The inverse function g^-1(x) is (1 - 3x)/(x). The values of x for which f(g(x)) = g^-1(x) are -9 ± sqrt(129)8. Send me the next one 📸