Confirm that f and g are inverses by showing the Composition of Inverses Rule for both f(g(x)) and g(f(x)) algebraically:
f(x)=9x+3
g(x)= (x-3)/9
thank you!
if y = 9 x + 3
reversing x and y yields
x = (y-3)/9
the other route
if y = (x-3)/9
reversing yields
x = (y-3)/9
9x = y - 3
or
y = 9 x + 3 sure enough
f(g) = 9g+3 = 9(x-3)/9+3 = x-3+3 = x
g(f) = (f-3)/9 = (9x+3-3)/9 = 9x/9 = x
That is the way to show that f and g are inverses
To confirm that f and g are inverses, we need to show that the composition of f(g(x)) and g(f(x)) equals x.
To find f(g(x)), we substitute g(x) into f(x):
f(g(x)) = 9(g(x)) + 3
Next, we substitute the expression for g(x) into f(g(x)):
f(g(x)) = 9((x-3)/9) + 3
Simplifying further:
f(g(x)) = x - 3 + 3
f(g(x)) = x
Therefore, f(g(x)) equals x.
Now, let's find g(f(x)):
g(f(x)) = (f(x) - 3)/9
Substitute the expression for f(x) into g(f(x)):
g(f(x)) = ((9x + 3) - 3)/9
Simplifying further:
g(f(x)) = 9x/9
g(f(x)) = x
Therefore, g(f(x)) also equals x.
Since both f(g(x)) and g(f(x)) equal x, we can confirm that f and g are inverses of each other.