Answer: a. To shift the graph of f(x) downward 5 units, we subtract 5 from the function: f(x) - 5. To shift it left 3 units, we replace x with x + 3: f(x + 3) - 5. To stretch it vertically by a factor of 4, we multiply the entire function by 4: 4[f(x + 3) - 5]. Finally, to reflect it about the x-axis, we take the negative of the function: -4[f(x + 3) - 5]. Therefore, the equation of the new function g(x) is:
g(x) = -4[1/2^(x+3) - 5]
b. To find the y-intercept, we set x = 0 in the equation of g(x):
g(0) = -4[1/2^(0+3) - 5] = -4[1/8 - 5] = -4[-39/8] = 195/2
Therefore, the y-intercept is (0, 195/2).
c. The domain of g(x) is all real numbers, since there are no restrictions on x.
d. To find the range of g(x), we first observe that the function is decreasing and asymptotic to y = -5 as x approaches infinity. This means that the range of g(x) is all real numbers less than or equal to -5.
Explanation: