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Let f(x) be a function with domain [−3,∞) and range (−∞,2). If g(x)= f^−1(x) what is the domain of g(x+4)−7?

Respuesta :

frika

Answer:

  • the domain of the function [tex]g(x+4)-7[/tex] is [tex](-\infty,-2);[/tex]
  • the range of the function [tex]g(x+4)-7[/tex] is [tex][-10,\infty).[/tex]

Step-by-step explanation:

If the function [tex]f(x)[/tex] has the domain [tex][-3,\infty)[/tex] and the range [tex](-\infty,2),[/tex] then the function [tex]g(x)=f^{-1}(x)[/tex] has the domain that id the function's [tex]f(x)[/tex] range and the range that is function's [tex]f(x)[/tex] domain. Therefore,

  • the domain of the function [tex]g(x)[/tex] is [tex](-\infty,2);[/tex]
  • the range of the function [tex]g(x)[/tex] is [tex][-3,\infty).[/tex]

The function [tex]g(x+4)-7[/tex] is the translation of the function [tex]g(x)[/tex] 4 units to the left and 7 units down, hence

  • the domain of the function [tex]g(x+4)-7[/tex] is [tex](-\infty,-2);[/tex]
  • the range of the function [tex]g(x+4)-7[/tex] is [tex][-10,\infty).[/tex]

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