Question

Consider the function g(x) =-4log (-1/3 x+2) +1.

a) Describe, in order and with detail, the transformations required to map f(x) -log(x) onto g b).
b) Find the equation of g⁻1 × (x).

Answer 1

To map f(x) = -log(x) onto g(x) = -4log(-1/3x+2) + 1, the transformations needed are a vertical stretch of 4, reflection about the x-axis, horizontal compression by a factor of 3, horizontal shift 2 units to the right, and vertical shift 1 unit up. To find g⁻¹(x), swap x and g(x) and solve for x.

Step-by-step explanation:

To map the function f(x) = -log(x) onto g(x) = -4log(-1/3x+2) + 1, the following transformations are required:

1. Vertical stretch of 4: This is achieved by multiplying the function by 4.
2. Reflection about the x-axis: This is achieved by multiplying the function by -1.
3. Horizontal compression by a factor of 3: This is achieved by multiplying the argument of the logarithm by 3.
4. Horizontal shift 2 units to the right: This is achieved by subtracting 2 from the argument of the logarithm.
5. Vertical shift 1 unit up: This is achieved by adding 1 to the entire function.

b) To find the equation of g⁻¹(x), we swap the roles of x and g(x) and solve for x. In this case, we have: g(x) = y = -4log(-1/3x+2) + 1. Swap x and y and solve for x.