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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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.`
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