# 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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