Two functions g and h are defined as :
g:x--> 2x+3/x-4
h:x --> 1/x
Write expressions for :
a) hg(x)
b) g inverse 1(x)
a) To find the expression for hg(x), we substitute the function g(x) into the function h(x).
hg(x) = h(g(x))
Substituting g(x) into h(x), we get:
hg(x) = h(2x + 3/x - 4)
b) To find the expression for g inverse 1(x), we need to find the inverse of the function g(x).
First, let's rewrite g(x) in equation form:
g(x) = (2x + 3)/(x - 4)
To find the inverse of g(x), we swap the x and y variables and solve for y:
x = (2y + 3)/(y - 4)
Multiply both sides by y - 4:
x(y - 4) = 2y + 3
Expand:
xy - 4x = 2y + 3
Rearrange the terms:
xy - 2y = 4x + 3
Factor out y:
y(x - 2) = 4x + 3
Divide both sides by (x - 2):
y = (4x + 3)/(x - 2)
Therefore, g inverse 1(x) is given by:
g inverse 1(x) = (4x + 3)/(x - 2)
To find the expression for hg(x), we need to substitute the function h into the function g.
a) hg(x) = g(h(x))
First, let's find h(x) by substituting x into the function h:
h(x) = 1/x
Now, we substitute h(x) into the function g:
hg(x) = g(h(x)) = g(1/x)
So, the expression for hg(x) is:
hg(x) = (2(1/x) + 3) / ((1/x) - 4)
= (2/x + 3) / (1/x - 4)
= (2 + 3x) / (x - 4x)
Now, let's find the expression for g inverse 1(x), which represents the inverse of the function g.
b) g inverse 1(x)
To find the inverse of g(x), we need to swap x and y in the equation and solve for y.
Let's start with the equation of g(x):
g(x) = (2x + 3) / (x - 4)
Swap x and y:
x = (2y + 3) / (y - 4)
Now, solve for y:
xy - 4x = 2y + 3
xy - 2y = 3 + 4x
y(x - 2) = 3 + 4x
y = (3 + 4x) / (x - 2)
So, the expression for g inverse 1(x) is:
g inverse 1(x) = (3 + 4x) / (x - 2)