SOLUTION: Let 𝑓 be the function with correspondence rule 𝑓(𝑥) = 3𝑥^2 + 5𝑥 − 3 with 𝑥 ∈ [0; +∞[, determine the inverse function 𝑓^−1 and its domain.

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Question 1194864: Let 𝑓 be the function with correspondence rule 𝑓(𝑥) = 3𝑥^2 + 5𝑥 − 3 with 𝑥 ∈ [0; +∞[, determine the inverse function 𝑓^−1 and its domain.
Found 2 solutions by shlomitg, MathTherapy:
Answer by shlomitg(8)   (Show Source): You can put this solution on YOUR website!
𝑓^−1(x)=1/(3𝑥^2 + 5𝑥 − 3) where 𝑥 ∈ [0; +∞] excluding xi which would lead to 1/0
Solving 3𝑥^2 + 5𝑥 − 3 = 0 we get x1=[-5-sqrt(61)]/6 x2=[-5+sqrt(61)]/6
x1=[-5-sqrt(61)]/6 <0 therefore not in the domain of the function
x2=[-5+sqrt(61)]/6>0 therefore in the domain of f(x), but not in the domain of 𝑓^−1(x) since it would lead to a value of 1/0
so the domain of 𝑓^−1(x) is 𝑥 ∈ [0; +∞] excluding [-5+sqrt(61)]/6]=0.468
Answer by MathTherapy(10555)   (Show Source): You can put this solution on YOUR website!
Let 𝑓 be the function with correspondence rule 𝑓(𝑥) = 3𝑥^2 + 5𝑥 − 3 with 𝑥 ∈ [0; +∞[, determine the inverse function 𝑓^−1 and its domain.
               , where:  
Use the formula: , to COMPLETE the SQUARE on this or ANY QUADRATIC.
            .
         
      
      Since   𝑥 ∈ [0, ∞), we opt for the "+" positive root, and so, we get: 

You need to use the RADICAND (expression UNDER the RADICAL) to determine the VALUES that are EXCLUDED as part of the INVERSE function's DOMAIN.
In this case, you can just use the NUMERATOR of the RADICAND, set it ≥ 0, and then solve the INEQUALITY to determine what RANGE of values are
EXCLUDED from the DOMAIN of the INVERSE function.
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