Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.
f(x) = (x-7)/(x+3) and g(x) = (-3x-7)/(x-1)
I got to the point where f(g(x)) is plugged in but how would I simplify it?
(((-3x-7)/(x-1))-7)/(((-3x-7)/(x-1)+3)
To simplify the expression f(g(x)), let's break it down step by step:
1. Substitute g(x) into f(x): Replace every instance of 'x' in f(x) with (-3x-7)/(x-1).
f(g(x)) = [(-3x-7)/(x-1) - 7] / [(-3x-7)/(x-1) + 3]
2. Simplify the numerator: Distribute the negative sign to both terms in the numerator.
f(g(x)) = [(-3x-7)/(x-1) - (7(x-1))/(x-1)] / [(-3x-7)/(x-1) + 3]
3. Combine like terms:
f(g(x)) = [(-3x-7 - 7x + 7)/(x-1)] / [(-3x-7)/(x-1) + 3]
4. Simplify the numerator further:
f(g(x)) = [(-10x)/(x-1)] / [(-3x-7)/(x-1) + 3]
At this point, you could simplify the expression further by finding a common denominator, but we can stop here because our goal is to show that f(g(x)) = x and not to completely simplify the expression.
Now, to continue and show that g(f(x)) = x, we'll follow the same steps, substituting f(x) into g(x):
1. Substitute f(x) into g(x): Replace every instance of 'x' in g(x) with (x-7)/(x+3).
g(f(x)) = [(-3(x-7)/(x+3)) - 7] / [(x-7)/(x+3) - 1]
2. Simplify the numerator: Distribute the negative sign to both terms in the numerator.
g(f(x)) = [(-3x + 21)/(x+3) - (7(x+3))/(x+3)] / [(x-7)/(x+3) - 1]
3. Combine like terms:
g(f(x)) = [(-3x + 21 - 7x - 21)/(x+3)] / [(x-7)/(x+3) - 1]
4. Simplify the numerator further:
g(f(x)) = [(-10x)/(x+3)] / [(x-7)/(x+3) - 1]
Again, we can stop here because our goal is to show that g(f(x)) = x and not to fully simplify the expression.
By showing that both f(g(x)) and g(f(x)) simplifies to x, we can conclude that f and g are inverses of each other.