b) Given the function f(t) = 1 - (3)/(t) and the range \-1, 3, 7 we need to find the domain.

Here are the solutions to your problems. b) Given the function f(t) = 1 - (3)/(t) and the range \-1, 3, 7\, we need to find the domain. The range consists of the output values of f(t). We set f(t) equal to each value in the range and solve for t. Step 1: Set f(t) = y and solve for t in terms of y. y = 1 - (3)/(t) y - 1 = -(3)/(t) (3)/(t) = 1 - y t = (3)/(1 - y) Step 2: Substitute each value from the range into the expression for t. For y = -1: t = (3)/(1 - (-1)) = (3)/(1 + 1) = (3)/(2) For y = 3: t = (3)/(1 - 3) = (3)/(-2) = -(3)/(2) For y = 7: t = (3)/(1 - 7) = (3)/(-6) = -(1)/(2) The domain is the set of these t values. The domain is \-(3)/(2), -(1)/(2), (3)/(2)\. a) Given a^x-3b^3x = a^5-xb^5x, we need to show that x ((a)/(b)) = 4 a. Step 1: Group terms with the same base on opposite sides of the equation. a^x-3a^5-x = b^5xb^3x Step 2: Apply the exponent rule (m^p)/(m^q) = m^p-q. a^(x-3) - (5-x) = b^5x - 3x a^x-3-5+x = b^2x a^2x-8 = b^2x Step 3: Take the logarithm of both sides. (a^2x-8) = (b^2x) Step 4: Apply the logarithm rule (M^p) = p M. (2x-8) a = 2x b Step 5: Distribute and rearrange the terms to isolate x. 2x a - 8 a = 2x b 2x a - 2x b = 8 a Step 6: Factor out 2x from the left side. 2x ( a - b) = 8 a Step 7: Apply the logarithm rule M - N = ((M)/(N)). 2x ((a)/(b)) = 8 a Step 8: Divide both sides by 2. x ((a)/(b)) = 4 a This completes the proof.