# a.) Show that h(x)=x^(3)/4 and k(x)=(4x)^(1/3) are inverses of one another. b.) Find the slopes of the tangents to the graphs at h an k at (2,2) and (-2,-2).THANK YOU SO MUCH!!

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### 1 Answer

a) You need to evaluate the inverse to `h(x)=(x^3)/4` to check if its equation coincides to equation of k(x) such that:

`h(x) = y => y = (x^3)/4 `

You need to write x in terms of y such that:

`4y = x^3 => x = root(3)(4y)`

Converting the cube root into a power yields:

`x = root(3)(4y) => x = (4y)^(1/3)`

`h^(-1)(x) = (4x)^(1/3)`

**Notice that equation `y = (4x)^(1/3)` coincides to the equation of the function `k(x) = (4x)^(1/3)` , hence, h(x) and k(x) are inverses of each other.**

b) You may find the slope of the tangent line to the graph of h(x), at the point (2,2), evaluating the derivative of h(x) at (2,2) such that:

`h'(x) = (3x^2)/4 => h'(2) = 3*4/4 => h'(2) = 3`

You may find the slope of the tangent line to the graph of k(x), at the point (-2,-2), evaluating the derivative of k(x) at (-2,-2) such that:

`k'(x) = (4/3)(4x)^(1/3-1) => k'(x) = (4/3)(4x)^(-2/3)`

`k'(-2) = (4/3)(-8)^(-2/3) => k'(-2) = 4/(3root(3)(8^2)) => k'(-2) = 4/(3*2^2) => k'(-2) = 1/3`

**Hence, evaluating the slope of the tangent line to the graph of h(x), at (2,2) yields `h'(2) = 3` and the slope of the tangent line to the graph of k(x), at (-2,-2) yields `k'(-2) = 1/3.` **