Consider the functions

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Consider the functions
f(x)= 5x+4/x+3(This is a fraction) and
g(x)= 3x-4/5-x(This is a fraction)

a)Find f(g(x))
b)Find g(f(x))
c)Determine whether the functions f and g are inverses of each other.

  • Consider the functions -

    f(g) = (5g+4)/(g+3)
    = (5(3x-4)/(5-x)+4) / ((3x-4)/(5-x)+3)
    = x

    g(f) = (3f-4)/(5-f)
    = (3((5x+4)/(x+3))-4) / (5-((5x+4)/(x+3)))
    = x

    since f(g) = g(f) = x, they are inverses

  • Consider the functions -

    What are the values that need to be excluded?

  • Consider the functions -

    whatever makes the denominator zero must be excluded, since division by zero is undefined.

    So, for f(g), x=5 is not allowed, since g(5) is not defined. In addition, since f(-3) is not defined, any x where g(x) = -3 must also be excluded. Luckily, there is no such x.

    Use similar reasoning for g(f).

  • Consider the functions -

    So there are no values to be excluded?

  • Consider the functions -

    Read what I said. You have to exclude x=5 because g(5) is not defined. Therefore, f(g(5)) is also not defined.

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