Question

The parent function `f\left(x\right)=\sqrt[3]{x}` is compressed vertically by a factor of `\frac{1}{3}` and then translated 3 units left and 7 units down. What is the transformed function `g\left(x\right)`?

Answer 1

The transformed function, after applying a vertical compression by a factor of ⅓, and translations 3 units left and 7 units down to the parent function f(x) = ∛x^3, is g(x) = ⅓ ∛(x + 3)^3 - 7.

Step-by-step explanation:

The transformed function g(x) results from applying a vertical compression by a factor of ⅓, a horizontal translation of 3 units to the left, and a vertical translation of 7 units down to the parent function f(x) = ∛x^3.

First, we apply the vertical compression by multiplying the function by ⅓:

• Compression: h(x) = ⅓ ∛x^3

Next, we translate the function 3 units to the left and 7 units down to obtain g(x):

• Horizontal & Vertical Translations: g(x) = ⅓ ∛(x + 3)^3 - 7

Therefore, the transformed function is g(x) = ⅓ ∛(x + 3)^3 - 7.

Answer 2

The answer is below

Step-by-step explanation:

Given that f(x) = x√3.

A function can be vertically stretched or compressed by multiplying it by a positive constant. If the constant is greater than 1, it is vertically stretched and if the constant is less than 1 it is vertically compressed.

If a function f(x) = x is compressed or stretched by a constant a, then the new function g(x) = a f(x)

If a function f(x) = x is translated a units down, then the new function g(x) = f(x) - a

If a function f(x) = x is translated a units left, then the new function g(x) = f(x-a)

If f(x) = x√3 is compressed vertically by a factor of 1/3. The new function is

`f(x)'=x√(3) *(1)/(3) \\\\f(x)'=(x)/(3) √(3)`

If it is then translated 3 units left and 7 units down, the transformed function g(x) is:

`g(x)=((x-3)/(3)√(3) )-7`