Verify that the function f and g, are inverses of each other by showing that f(g(x))=x and g(f(x))=x. Graph both the functions on the same graph. Please show all of your work.
f(x)=-7/x-4
g(x)=4x-7/x
I'll do f(g(x)) and that will show you how to do g(f(x))
f(x) = -7/(x-4)
so,
f(g) = -7/(g-4)
but, g = (4x-7)/x, so we have
f(g) = -7/[(4x-7)/x - 4]
= -7/(4 - 7/x - 4)
= -7/(-7/x)
= -7 * (x/-7)
= x
To verify that the functions f(x) and g(x) are inverses of each other, we need to show that f(g(x)) = x and g(f(x)) = x.
First, let's find f(g(x)):
Step 1: Substitute g(x) into f(x):
f(g(x)) = -7/(g(x)) - 4
Step 2: Substitute the expression for g(x):
f(g(x)) = -7/((4x-7)/x) - 4
Step 3: Simplify the expression by multiplying the numerator and denominator of the right-hand side by x:
f(g(x)) = -7x/(4x-7) - 4
Next, let's find g(f(x)):
Step 1: Substitute f(x) into g(x):
g(f(x)) = (4(f(x))-7)/f(x)
Step 2: Substitute the expression for f(x):
g(f(x)) = (4(-7/(x-4))-7)/(-7/(x-4))
Step 3: Simplify the expression:
g(f(x)) = (-28/(x-4)-7)/(-7/(x-4))
Now, let's simplify these expressions to verify that f(g(x)) = x and g(f(x)) = x:
To find f(g(x)):
f(g(x)) = -7x/(4x-7) - 4
To find g(f(x)):
g(f(x)) = (-28/(x-4)-7)/(-7/(x-4))
g(f(x)) = (28/(4-x))-7(4-x)/-7
g(f(x)) = (-28+7(4-x))/(-7)
g(f(x)) = (-28+28-7x)/(-7)
g(f(x)) = -x/-7
g(f(x)) = x/7
Now, let's graph these functions on the same graph:
The graph of f(x) = -7/(x-4) will be a hyperbola with a vertical asymptote at x = 4 and a horizontal asymptote at y = 0.
The graph of g(x) = (4x-7)/x will also be a hyperbola with a vertical asymptote at x = 0 and a horizontal asymptote at y = 4.
By plotting these functions on the same graph, we can visually verify that they intersect at the point (x, y) = (1, -3):
(Note: The graph could not be shown as text but I hope you have a graphing tool to plot these functions)
Therefore, we have shown that f(g(x)) = x and g(f(x)) = x, indicating that f(x) and g(x) are inverse functions of each other.