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.
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
What are the values that need to be excluded?
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).
So there are no values to be excluded?
Read what I said. You have to exclude x=5 because g(5) is not defined. Therefore, f(g(5)) is also not defined.
To find the composition of two functions, f(g(x)) and g(f(x)), we substitute the expression of one function into the other.
a) To find f(g(x)), we replace every occurrence of x in f(x) with g(x):
f(g(x)) = 5g(x) + 4 / g(x) + 3
Now, let's find g(x):
g(x) = (3x - 4) / (5 - x)
Replacing g(x) in f(g(x)):
f(g(x)) = 5[(3x - 4) / (5 - x)] + 4 / [(3x - 4) / (5 - x)] + 3
Now, simplify the expression. Multiply both the numerator and denominator of the first fraction by (5 - x) to eliminate the complex fraction:
f(g(x)) = 5(3x - 4) + 4(5 - x) / (3x - 4) + 3(5 - x)
Simplifying further, we get:
f(g(x)) = (15x - 20 + 20 - 4x) / (3x - 4 + 15 - 3x)
= (11x) / (11)
= x
Hence, f(g(x)) = x.
b) To find g(f(x)), we replace every occurrence of x in g(x) with f(x):
g(f(x)) = (3f(x) - 4) / (5 - f(x))
Now, let's find f(x):
f(x) = (5x + 4) / (x + 3)
Replacing f(x) in g(f(x)):
g(f(x)) = (3[(5x + 4) / (x + 3)] - 4) / (5 - [(5x + 4) / (x + 3)])
Now, simplify the expression. Multiply both the numerator and denominator by (x + 3) to eliminate the complex fraction:
g(f(x)) = (3(5x + 4) - 4(x + 3)) / (5(x + 3) - (5x + 4))
Expanding the brackets:
g(f(x)) = (15x + 12 - 4x - 12) / (5x + 15 - 5x - 4)
Simplifying further, we get:
g(f(x)) = (11x) / (11)
= x
Hence, g(f(x)) = x.
c) To determine whether the functions f and g are inverses of each other, we need to check if their composite functions f(g(x)) and g(f(x)) are equal to the identity function, f(x) = g(x) = x.
From the previous calculations, we found that both f(g(x)) and g(f(x)) are equal to x. Hence, the functions f and g are inverses of each other.