9514 1404 393
Answer:
f⁻¹(138) = 3
Explanation:
You want to find the value of x that makes the function have a value of 138:
f(x) = 5x^3 +3x -6
138 = 5x^3 +3x -6
0 = 5x^3 +3x -144
Descartes's rule of signs tells us this has one positive real solution. The rational root theorem gives us 30 possibilities. Rewriting the equation as ...
x^3 = (144 -3x)/5 = 28.8 -0.6x
suggests that the value of x is less than ∛28.8 ≈ 3.065. Trying x=3, we find that to be a solution.
(5x² +3)(x) -6 = 0 . . . . rewrite of the above equation
(5·3² +3)·3 -144 = (48)(3) -144 = 0 . . . . true
Then ...
f⁻¹(138) = 3
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The answer is found easily using a graphing calculator. The solution is the x-intercept of 138 -f(x) = 0.