Help me find inverse. f(x)=4x^3-5,X=set of real numbers x = 4y3 – 5 x+5 = 4y3 (x+5)/4 = y3 ∛((x+5)/4)= y?
This is what I have so far...
f(x)=4x^3 - 5, X=set of real numbers
x = 4y^3 – 5
x+5 = 4y^3
(x+5)/4 = y^3
∛((x+5)/4)= y
I feel like I should reduce this cube root more and get the fraction out from underneath the radical sign but I'm unsure.
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No you're fined because the only chance to ∛((x+5)/4) would be ∛(x+5)/∛4 based on (a/b)^n = a^n/b^m and that not really a simplification though can be a useful step sometimes
f(x) = 4x^3 - 5
=> f(f⁻¹(x)) = 4(f⁻¹(x))^3 - 5
=> x = 4(f⁻¹(x))^3 - 5
=> x + 5 = 4(f⁻¹(x))^3
=> (x + 5)/4 = (f⁻¹(x))^3
=> f⁻¹(x) = ((x + 5)/4)^(1/3)
=> f⁻¹(x) = ∛((x+5)/4)